The rules of the FA Cup permit five substitutions and Carlo Ancelotti used his full complement.

Abdoulaye Doucoure was one of the first introduced, though, brought on with Bernard to inject energy and thrust as Ancelotti retaliated to Rotherham's equaliser five minutes earlier.

Doucoure's drive and combativeness gradually changed the complexion of Everton's football, which had lost its way after a storming start which resulted in them going ahead before Rotherham had got out their own half.

And Doucoure eventually had the decisive say, although the goal owed a lot to the vision and guile of James Rodriguez.

Collecting possession from fellow Colombian Yerry Mina, James spied Doucoure advancing between two defenders and supplied a precise pass, inviting the midfielder to continue forwards and apply a crisp left-foot finish and finally subdue the game Championship team.

Cenk Tosun had floated home a beautiful chip for the opener, a first Everton goal in 433 days for the striker.

Matt Olosunde, Rotherham's bright American wing-back, formerly with Manchester United, brought the Cup tie back to the boil with a well-placed equaliser 11 minutes after half-time.

But the combination of James and Doucoure, both playing for Everton in the FA Cup for the first time – and some excellent goalkeeping from Robin Olsen – booked a fourth-round spot for manager Ancelotti's team.

The old trope concerning these games which pit Premier League class against a definite underdog concerns the value of an early goal.

It is oft recycled because for the most part it’s true.

Get on top early and sap the spirit of the lower division hopefuls.

As it happened, going behind seemed to breathe life into a Rotherham team whose initial objectives seemed to belong in the categories of denying and surviving.

Everton’s breakthrough arrived on nine minutes and even with the game so young, it had been coming.

Seamus Coleman crashed through a challenge on Jamie Lindsay to signal Everton’s intent and the home team moved the ball quickly and progressively.

Anthony Gordon rolled Michael Ihiekwe on the halfway line after collecting a pass from James – occupying a number 10 role but routinely appearing deep in the opening minutes – and advanced into the box for a shot blocked by Ben Wiles.

Andre Gomes, one of eight changes from the side which faced West Ham United on New Year’s Day, scuffed wide after a clearing header from Angus MacDonald.

Everton were in complete control. A statistic after eight minutes showed they’d had 89 percent of possession.

The imposing MacDonald was one of three Rotherham centre-halves, flanked by originally defensively-minded wing-backs and protected by a four-strong midfield.

That trio of centre-backs wasn’t on the same page, however, when Gordon escaped Ihiekwe for a second time and progressed infield to slide through a pass for Tosun.

Richard Wood was lagging behind his colleagues, playing Tosun onside.

The striker, starting an Everton match for the first time in 413 days, strode into the penalty area and conjured a finish which conclusively denied his recent relative inactivity.

Confronted by goalkeeper Jamal Blackman, advancing from his line, Tosun produced one of those gorgeous chipped efforts which sees the ball pleasingly amble into the net.

Tom Davies nearly connected with Tosun when attempting a ball from deep in his own half not long after.

Then for no apparent reason other than Rotherham seemingly decide to have a right go, the tide turned.

Ihiekwe snatched at a ball bouncing in the box to shoot over the bar.

Michael Smith struck with more conviction but the same result after Matt Crooks burst to the edge of the box and supplied a square pass.

Right wing-back Olosunde, formerly restricted to trying to contain the twin threat of Lucas Digne and Gordon on Everton’s left, arrived as an attacking force on 25 minutes.

Olosunde was completely emboldened by the time he levelled – not long after tentatively asking for a penalty when falling with Gomes at his back – but his first contribution of note was a driving run into the box.

Olsen used his legs to stop the shot, with Michael Keane doing likewise to repel Dan Barlaser’s follow up.

It was Lindsay’s turn next, as Rotherham sustained this assault on Everton’s goal, Olsen taking off to touch over a rising 20-yard drive.

Olsen used his fist to clear when Ihiekwe threatened to fasten onto a flick from Smith – while at the other end, Coleman, who persistently tried to create on Everton’s right, swapped passes with Alex Iwobi but saw his low cross turned wide of the far post by Tosun.

Iwobi flashed an effort off target shortly before Olosunde drew Rotherham level.

There was nothing particularly dangerous about the play unfolding in the build-up to Olosunde’s strike.

Smith turned a bouncing ball forwards with his head, Davies first to it in a bid to tidy up.

Davies’ touch steered the ball right for Olosunde, however, and he slid in to squeeze the ball inside Olsen’s right post.

Olosunde on the front foot again not long after, sliding a pass across goal for fellow wing-back Wes Harding, whose strike lacked composure and flew over.

Meantime, James and Gomes both had optimistic strikes blocked by defensive bodies.

And following a wave of substitutions, James’ skidding effort from distance deflected behind for Everton’s first corner of the game on 72 minutes.

Those Everton changes – Bernard, Doucoure, Mina and Gylfi Sigurdsson all on in short order – meant a switch to left-back for Ben Godfrey, who had started in his favoured centre-back position.

For Rotherham George Hirst came on and within moments would likely have scored but for Michael Keane recovering to apply a vital toe to the ball.

There was another close-run thing for Hirst with five minutes remaining.

The striker darted in front of Keane to meet a cross from the left but his contact was negligible and only directed the ball away from goal.

Everton, though, were steadily regaining some form of offensive momentum.

Blackman did very well to save point blank from Tosun after Mina steered Wood’s skied clearing header back into the danger zone.

Tosun was convinced he’d settled the tie when beating Blackman to James’ whipped free-kick from the left and toeing home.

The VAR, though, drew his lines and concluded it should be scrubbed out for offside. It was desperately close.

Ultimately, the video intervention, on 86 minutes, didn’t matter.

Doucoure had grabbed the game by the scruff of the neck after coming on to add substance to midfield.

The Frenchman was positive and forceful. And when he found space between MacDonald and Wood three minutes into extra time, it would prove the game's decisive episode.

James’ ball split the defenders, enabling Doucoure to take the pass in his stride and bury emphatically beyond Blackman for his second Everton goal.

That blow for Rotherham set the away side back on their heels.

Niels Nkounkou immediately came on for James and sent in a cross which Bernard guided against the right post.

Blackman saved from Coleman and Tosun was fractions from connecting with another Nkounkou delivery.

Rotherham gave it one final push. 

The game was 114 minutes old when Mina – now with a bandage wrapped round his head after a prolonged stoppage for treatment – threw his body in the way of a goalbound attempt from substitute Kyle Vassell.

And, eventually, the day's sole mission, FA Cup progress was accomplished.

Cenk In Tune

Nine minutes. That was all Cenk Tosun had to find his bearings before being presented with a chance to hand Everton a precious leg-up against a Rotherham team initially set up to frustrate and thwart.

Tosun’s competitive football this season amounted to 45 minutes, spread across six substitute appearances, following a return from injury on 1 November.

The Turkey striker last started a match for Everton 413 days ago, on 23 November 2019.

This was the first time he’d begun an FA Cup tie.

Tosun was stepping into the gigantic shoes of Dominic Calvert-Lewin, to boot.

The position at the front of an attack can be a lonely one, even when your team is overwhelmingly on top.

There is a lot of selfless work attached to the job and patience is a necessary virtue.

Opportunities come at a premium and a forward stands or falls on what he does with them.

Tosun read the play when Anthony Gordon gathered possession on the left.

Gordon had left Michael Ihiekwe, Rotherham’s right-sided centre-back for dead once already.

Tosun backed his teammate to repeat the feat and was rewarded for his confidence when Gordon escaped to thread an immaculate ball through the middle.

It was all on the centre-forward, now, advancing towards goal with keeper Jamal Blackman closing to narrow the angle.

This was the point at which any rustiness would have been exposed.

Instead, Tosun delicately swept his left boot through the ball, caressing a skilful finish over Blackman.

It was the 29-year-old’s first Everton goal since 3 November 2019. He deserved his celebration.

Tosun’s next moment of jubilation was cut short by the VAR.

It was an indication of the player’s sharpness and endurance nonetheless that he clinically turned home from James Rodriguez’s free-kick with four minutes remaining, only for an offside decision to go against him.

He was close to another in extra-time, shooting a yard wide, but Tosun can be very pleased with the 120 minutes of football in his legs and a decisive contribution to his side's victory.

Olsen Stands Firm To Weather First-Half Millers Storm

The impression created by Rotherham at the start of this was one of a team prepared to hang back and defend.

And should anything fall in their favour offensively, well that would be a bonus.

The away team had five players strung across the back and four midfielders in close contact ahead of them.

This kind of approach doesn’t typically alter after the concession of one goal, the team operating on the back foot generally content to stick in the game with only one goal to recover.

Around the midpoint of the opening half, however, Rotherham injected intent and ambition previously unseen.

The transformation was perhaps epitomised when Robin Olsen had to scarper from his line to punch clear from Michael Ihiekwe, the furthest Rotherham player forward despite being a member of the visitors’ back three.

That was just past the half hour and by this stage Olsen well and truly had his eye in.

Initial warnings of Rotherham’s renewed gusto came from Ihiekwe and Michael Smith.

The former spun and hammered a bouncing ball over the top on 21 minutes.

Four minutes later, striker Smith was high with a vicious blast from 18 yards.

Olsen’s first test came soon after, the Swedish goalkeeper using his legs to block Matt Olosunde’s low effort following a gallop into the box from the right wing-back.

Michael Keane stood tall to deny the first follow up from Dan Barlaser.

But when the ball was recycled for Jamie Lindsay to unload from 20 yards Olsen produced his best yet, meeting the rising ball with his fingertips to concede a corner.

The linesman’s flag was already up when Olsen tumbled to his right to keep out a subsequent Lindsay volley but the save was indicative of a player in the groove.

There was a reaction stop to deny a near-post header that – it transpired – wouldn’t have counted either.

This was a key period in the game nonetheless, the Championship side fearless and sensing opportunity.

Everton and Olsen in particular were on their mettle to survive a dicey period.

The Everton keeper, on loan from Roma, was eventually beaten on 56 minutes and there was nothing he could do to stop Olosunde’s effort sneaking inside his far post.

That Olosunde’s goal only brought Rotherham level, meaning Everton could go and win the game in extra time, owed a lot to Olsen’s first-half performance.

Carlo Ancelotti is progressively compiling a competitive Everton squad and the goalkeeping position is in a few very safe pairs of hands indeed.

First Hurdle Cleared

Everton will learn both their fourth-round opponents and potential opposition for round five on Monday.

The double draw is a product of this congested and unique season – with Everton back in FA Cup action in a fortnight.

There was no looking beyond the task in hand, here. 

In the same way golfers talk about not being able to win a major in the opening two rounds, only lose it, no FA Cups were ever won on third-round day.

The task for Carlo Ancelotti’s team on Saturday was prosaic in nature and, to borrow a phrase from the Italian manager, one which required them to pay attention.

Additionally, Everton have had a strained relationship with the old competition of late.

In four of the past six seasons, they’ve not gone past this stage. On the two other occasions, they reached one semi-final and suffered a defeat in round four.

Rotherham were liberated from their battle to remain in the Championship, the result having no bearing on their wider objective.

An opponent with nothing to lose is, by definition, very dangerous. Particularly one chasing the carrot of a day in the sun.

Everton started quickly, hogging possession and moving the ball quickly and precisely.

They got their noses in front after nine minutes.

But Cenk Tosun’s opener served to tweak the tiger’s tail, Rotherham rallying and deserving their shot at an extra half-hour.

Equally, Ancelotti’s side took complete charge in the added period, scoring early and meriting their victory.

Everton didn’t win the FA Cup today – but nor did they lose it and that was all that could be asked.