Carlo Ancelotti vowed Manchester City would leave Goodison Park on Saturday knowing they’d been in a game.
And the Italian’s team ensured their manager was true to his word.
Ancelotti employed a gameplan similar to the one that defeated Liverpool last month and saw his side similarly restrict their opponents.
Manchester City had an awful lot of the ball but it was 14 minutes before goalkeeper Joao Virginia – on his second Everton start – had a meaningful touch.
Virginia brilliantly denied Raheem Sterling after half-time but was otherwise relatively untroubled.
Everton, who began the game suffocating and rattling City, increasingly posed a threat of their own.
Richarlison was marginally high with an overhead kick and created a number of promising situations.
The hosts began to drop deeper with around 20 minutes remaining, a product of energy spent pressing and hounding Pep Guardiola’s players.
Ayemric Laporte exploited the extra space to stride forwards on 84 minutes, He fed Kevin De Bruyne, who tried to evade Yerry Mina but inadvertently shovelled the ball back into the path of centre-back Laporte.
Virginia pushed the ball onto the bar but Ilkay Gundogan headed the rebound into an empty net.
Forced to chase an equaliser, Everton left holes at the back for the first time and De Bruyne raced into one of them to complete the scoring on 90 minutes and end the hosts' FA Cup ambitions at the quarter-final stage.
It was City's 25th win in 26 matches and not many will have been as hard earned as this.
Everton have a fortnight off, a rare commodity in this peculiarly congested season, before returning to purse their overarching target for the campaign: European qualification.
Everton called City’s bluff to create their first chance – although to call it a chance is to acknowledge the ability of Richarlison to conjure something out of the ordinary.
Mason Holgate had unsettled the away side with an earlier throw-in delivered flat into the box.
Expecting the same, City backed off, only for Holgate to go short to Allan.
His cross reached Ben Godfrey at the back post, helping the ball in to the middle once, then twice, after Ruben Dias’ partial clearance.
The second delivery dropped in the vicinity of Richarlison, who had his back to goal.
Undeterred, the Everton striker improvised, contorting his body to make strong contact with an overhead kick that sent the ball narrowly over.
More reasons for Everton encouragement a quarter-hour later – in the 34th minute.
Everton had to use their possession wisely, counselled Ancelotti on the eve of this game.
The passing, insisted Everton’s manager, must be fast and vertical.
His players were listening: Andre Gomes forward, into Gylfi Sigurdsson, rapidly going in the same direction for Dominic Calvert-Lewin.
Zack Steffen, Manchester City’s goalkeeper, sped from his penalty area, to the annoyance of Laporte, who thought he had the situation in hand.
Keeper and defender got in a tangle, leaving Calvert-Lewin keeping his fingers crossed for a favourable bounce.
He didn’t get it and City were able to smuggle the ball behind.
Oleksandr Zinchenko, hitherto mainly seen as an attacking outlet, rescued his side 60 seconds before half-time.
Everton, whose confidence was climbing with every neatly-constructed advance, came forwards via Sigurdsson and Richarlison, eventually winning a corner.
Lucas Digne took it, from the right, finding the forehead of Mina, whose glancing header was prevented from creeping inside the far post by Zinchenko, positioned on the line.
City ended the half with an effort from Phil Foden – their best player – which deflected off Godfrey and careered awkwardly over the top.
And for all their possession in the opening 45 minutes – 77 per cent of it – that was typical of the away team’s success in Everton territory.
Ancelotti’s side smothered and hounded, plugged gaps and, in Godfrey, had a speed merchant who was a match for the express Sterling.
The first task was to prevent City from settling.
Calvert-Lewin swarmed all over Fernandinho, Godfrey and Seamus Coleman were both on top of Foden when he pulled the trigger from 20 yards and Holgate slid to prevent a pass reaching Sterling in a dangerous position.
Everton were still at it on 25 minutes, Sigurdsson and Allan ganging up on Gundogan and forcing the irked German to pass backwards.
There was one rare lapse from Coleman, surrendering the ball to Gabriel Jesus.
Everton’s captain was evidently very put out by this turn of events and hassled the Brazilian until he regained possession and left a fed up Jesus flat on the turf, into the bargain.
It was an episode the epitomised Everton’s relentless first-half effort.
When Allan was penalised for a challenge on Fernandinho 35 yards out, Foden spied an opportunity to examine Virginia’s nerve.
The keeper wasn’t’ required, Foden accurate with his strike but Mina getting his head in the line of the ball.
When Gundogan ran into Gomes and Gundogan on Everton’s 18-yard line after 14 minutes, the hosts tried to counter.
Gomes strode forwards but his pass for Richarlison was slightly overhit. Ancelotti glanced plaintively at the sky but he’d anticipated Everton having no margin for error on the ball.
The lucky few allowed inside Goodison Park gasped like a pantomime audience when Holgate appeared to clip Foden in the penalty area – the City midfielder remaining on his feet following the minimal contact.
When City finally shot on target, Virginia was equal to Sterling’s well-struck attempt – the away player directing a free-kick too high soon after.
On one occasion, Everton appeared to go against their manager’s advice, slowing up an attack because Sigurdsson saw no passing options.
He waited and, eventually sprung a ball to Gomes wide on the left.
The first-time cross was fabulous and would have landed on Calvert-Lewin’s head but for Laporte’s canny positioning.
Richarlison, a menace whenever Everton attacked, was wide from distance – after compatriot Allan went into the book for a challenge on Gundogan, which brought a very loud response from the City contingent.
Fernandinho took advantage of an Allan slip to progress through the middle of the field six minutes after the restart.
He evaded Mina’s challenge in the penalty-box ‘D’ but the pass forced Sterling wide and Virginia gobbled up the England player’s low cross.
Fernandinho received a yellow card six minutes later, punished for a crude barge on Richarlison in City’s half.
Sigurdsson lofted the free-kick into the box and when the ball ricocheted off Mina for Richarlison, the South American’s jabbed, right-foot strike travelled narrowly past the post.
City duly returned fire, Fernandinho again, suddenly involved at every turn and receiving a pass from Sterling on Everton’s right.
The delivery was along the floor and found Sterling, who had continued his run into the middle.
Sterling’s shot wasn’t clean, rearing off the surface, but the save from Virginia to prevent the ball sneaking inside his left post was outstanding.
The real shame of this blockbuster tie, of course, was the absence of supporters.
And how Goodison would have rocked and roared its approval when Godfrey burst from his own penalty area to execute a blood-and-thunder challenge on Foden.
It was a replica of the excellent Everton defender’s tackle on Liverpool’s Xherdan Shaqiri last month.
No stopping Foden having a crack from 20 yards with 24 minutes remaining, the ball flashing beyond Virginia’s right post.
Gomes got his angles wrong with a venomous shot after a Sigurdsson corner was partially cleared, then a VAR check determined Dias was blameless when struck on the arm by a Richarlison cross.
A booking for Gomes meant two Everton players treading a disciplinary tightrope and it was heart-in-mouth time when either the Portuguese or Allan tried to win the ball.
City unquestionably knew the identity of the home players who’d got on the wrong side of referee Michael Oliver, too.
Virginia got down snappily to hold another Foden blast – and did exceptionally well to tip Laporte’s rising strike onto the meat of the crossbar.
The rebound could have gone anywhere and Gundogan counted his blessings when it ran into the midfielder’s path.
He stopped to head into an unguarded net and inflict what would be a a fatal blow on Everton’s FA Cup hopes.
Everton rallied, Gomes releasing Calvert-Lewin, whose attempt with two minutes left deflected and squirmed across the box.
And City secured their FA Cup progress when Rodri funnelled the ball forwards for fellow substitute De Bruyne, who manoeuvred the ball onto his left foot for an emphatic strike beyond Virginia.
It felt very hard on the home team.
