For six minutes, Carlo Ancelotti’s first substitution appeared to have swung a frantic game, decisively, in Everton’s favour.
Seamus Coleman replaced Alex Iwobi on 61 minutes and 60 seconds later, after collecting a header from Tom Davies, took off down the right.
The Irishman combined with Richarlison to play around Eric Dier, then retained his composure to locate the one blue shirt in the box.
It belonged to Gyfli Sigurdsson, already on the scoresheet after dispatching a penalty in response to Harry Kane’s opening goal.
Sigurdsson opened up his body, allowing the ball to travel across him, before jabbing out a solid left foot to score past Hugo Lloris.
It was an exceptional finish to an invigorating passage of play, characteristic of the bustling, forceful Coleman.
Tottenham were second best for large parts of this match and it seemed we would have a result to reflect that fact.
In Kane, however, the Londoners boast an enduring threat.
Erik Lamela came on for Tottenham and struck a 68th-minute right-wing cross to the far post, where Kane capitalised on confusion in the hosts’ defence to crack in the equaliser.
Another Everton substitute in Joshua King was denied one-on-one by Lloris close to the end, Richarlison off target from a promising position with the follow up.
Everton, then, remain eighth, three points off the top six and three further back from the top four.
They go to Spurs' capital rivals Arsenal in seven days very clear on the required outcome.
It took going behind for Everton to locate their rhythm and launch what could fairly be described as an assault Tottenham’s goal.
Lloris enjoyed a quiet time of it for the opening half hour, not needed for anything other than a routine save down to his left after James Rodriguez slid a ball inside Toby Alderweireld for Richarlison to gather and shoot.
James, inevitably, was involved again when Everton scored their first goal.
The starring role, however, belonged to Sigurdsson.
He collected a pass from Michael Keane – that pair combining repeatedly, Sigurdsson coming off his left flank to receive very good balls from the centre-half – and galloped down the left.
Arriving parallel to the 18-yard line, Sigurdsson slowed up, processing the scene, then rolling a square ball for James.
The Colombian made to shoot, drawing back his left boot, but was prevented from completing the act by former Real Madrid teammate Sergio Reguilon.
Illegally, in the view of Michael Oliver, the referee.
Sigurdsson’s penalty was passed into the bottom right corner and impossible to stop.
Allan, back in Everton’s midfield after four weeks out, was prompting and intercepting, guarding his back four and quickening the transition from defence to attack.
When the Brazilian picked up Reguilon’s lax clearance, the ball forward for James was excellent.
James’ back-heeled flick was better still, feeding Sigurdsson, who was on the same wavelength as his gifted colleague, feeding a return pass.
Lloris tumbled left, once more, to make the save.
But Everton had regained control.
Playing with a back-four, albeit reverting to three central defenders on the rare occasions they were forced to defend, the hosts had tempo and intensity.
Between the equaliser and Lloris’ save from James, there was a Sigurdsson free-kick from 20 yards into the wall and a James long ranger watched all the way into his gloves by the Tottenham goalkeeper.
Tottenham had four attempts in the opening half, only one on target.
That was all they needed to hit the front.
Ben Godfrey and Keane combined to block a thumping Tanguy Ndombele effort but the ball was recycled and eventually funnelled, via Reguilon, to French midfielder Ndombele on the left.
Keane got a head on the inswinging cross but could only help it along its trajectory towards Kane, profiting from Mason Holgate’s slip to stand in space at the back post.
The finish was everything you’d expect from a man reaching 20 goals in a Premier League season for the fifth time – keeping company with Alan Shearer, Sergio Aguero and Thierry Henry.
He controlled and rifled left footed inside the far post. It was clinical and, for Everton, hurtful.
The response, however, was compelling.
Richarlison had a strike deflected over and from Sigurdsson’s resultant corner leapt above Alderweireld but aimed directly at Lloris.
Lucas Digne defended brilliantly at the back post to prevent Pierre-Emile Hojbjerg connecting with a Dier cross shortly after the restart – and following Richarlison heading over from a corner.
Jordan Pickford, returning from the abdominal injury which kept him out for five weeks, saved from Son Heung-min at his near post after the Korean jinked into the area on Tottenham’s left.
Everton had begun the game with intent. Joe Rodon jumped to prevent Allan’s pass setting Richarlison free.
A fluid sequence of passes, starting when Allan dispossessed Kane, and continuing with Davies transferring the ball through the air for Richalison, ended with Sigurdsson striking over.
The traffic continued flowing, primarily, toward the visitors' goal, save for the setback delivered by Kane.
Spurs were hanging on for the interval – and grateful Godfrey couldn’t control his effort after steaming in to meet a James free-kick allowed to travel a long way, left to right.
Davies’ 55th-minute sliding tackle on the breaking Reguilon, as the away side livened up, was immaculately timed – and it had to be, the Everton midfielder in the book after arriving late into a first-half challenge on the busy Ndombele.
Ndombele again, striding into Everton’s box for a shot that cannoned off Digne and spun over the bar.
Alderweireld met the corner with a header which smacked Pickford’s left post.
If the chances were being shared more equally in the second half, then it was Everton still looking most likely.
Richarlison had the ball in the net but was offside when cushioning Allan’s ball over the top.
The disappointment was short-lived, Sigurdsson emphatically converting soon after.
But for Coleman, read Lamela, a player introduced and having an immediate impact for his team.
The Argentine didn’t appear to have anywhere to go, in a spot on Everton’s left.
He eventually dug out a cross, nonetheless, Kane receiving the ball close to goal, unmarked for a second time.
The blast beyond Pickford was the equal of the England players’ first goal for both its ruthlessness and inevitability.
A header from Kane to complete the hat-trick seemed to take an eternity to pass Pickford’s left post, the keeper frantically trying to cover the ground.
Lamela was too high from distance as the game opened up, both sides recognising the value of two additional points.
With five minutes remaining, it looked, for one teasing moment, as if Anceotti had done it again.
King was barely on for Coleman than he was released by a subtle James pass.
Lloris saved with his legs but the rebound feel for Richarlison, who began the move with a fine ball for James.
The forward snatched at his shot, landing a frustrated boot on the advertising hoarding behind the goal after skewing over.
James had a half-chance late-on, too, but was off balance and off target after being fed by Sigurdsson 12 yards out.
Everton have it all to do - but, crucially, all still to play for in their European fight.
Sigurdsson's Captain's Effort
When Everton needed pulling up by their bootstraps it was captain Gylfi Sigurdsson tugging hardest.
The Icelander clearly savours these encounters against his old employers.
During the barmy FA Cup tie two months ago – won 5-4 by Everton – Sigurdsson scored once and was responsible for the creation of another three goals.
To illustrate the rarity of that feat, the previous Everton player to claim a hat-trick of assists was Steven Pienaar nine years earlier, in 2012.
Sigurdsson has stood up when his team’s needed him on multiple occasions this term.
There was a cool-as-ice penalty to defeat Chelsea and late strike to beat Sheffield United, those efforts bookending a four-match winning Premier League run.
What about the nonchalant finish from the spot to sink Liverpool at Anfield?
And, here again, Sigurdsson took charge when his team’s hopes – on the night and for the season – hung in the balance.
There had been nothing in this for Spurs when Harry Kane opened the scoring with the visitors’ first shot on target.
And, momentarily, even with no fans in the stadium, you could feel the oxygen escaping the place.
Sigurdsson, however, breathed fresh air into Goodison with a charge down the left and accurate ball across the fringe of the area, locating James Rodrigruez, who was upended by Sergio Reguilon.
The penalty was nervelessly placed in the bottom corner for Sigurdsson’s seventh goal of the season.
Number eight was a cracker, Sigurdsson’s execution of the highest technical order to redirect Seamus Coleman’s cross beyond the helpless Lloris.
This was a captain’s performance from Sigurdsson, ensuring his side didn’t finish the night empty handed.
Allan Shows Value
Allan audibly upbraided himself after defender Joe Rodon intercepted a through ball on 10 minutes.
Everton’s midfield enforcer was back from four weeks off and keen to make an impression.
The Brazilian’s manager, Carlo Ancelotti, prefers to keep his counsel over injury frustrations.
Surely, though, Ancelotti will have his fingers crossed for a long unbroken run for the player he bought from old club Napoli on the eve of this campaign.
Allan was making only a 16th Premier League start, here, after his season was twice interrupted by injury.
The importance of the player was underlined when he came back against Southampton on the first day of March.
Instrumental in a 1-0 win, Allan was rested for a trip to West Bromwich Albion three days later.
Summoned from the bench with that game meandering towards a scoreless draw, he provided balance and bite, a platform for Everton to claim three points.
Harry Kane found Allan blocking his path when carrying the ball infield and the South American was soon running over Tanguy Ndombele to claim midfield possession.
Allan’s positional sense guides him into spaces to collect loose balls, a quality in evidence when Sergio Reguilon’s loose clearance was picked up by Allan 50 yards from goal.
It looks like a stroke of fortune but it happens too often to be entirely down to luck.
Allan’s subsequent pass to James Rodriguez was crisp and accurate, loaded with sufficient weight for James to lay-off first time.
It was a ball typical of Allan, who hurries his passes in a forward direction, along the floor.
A lot of the 30-year-old’s work goes unnoticed; a toe on the ball to stop Ndombele escaping down the left, for example.
Followed in quick succession by a sprint to close down Pierre-Emile Hojbjerg and force the Spurs player to steer a pass into touch.
That one was noisily greeted by Everton’s coaching staff.
Allan’s quick feet enable manipulation of the ball in tight areas – a product of a futsal upbringing – and he can generate the impression of a bull fighter teasing his foe.
A pass flipped over the top for Richarlison provided another window into what Allan can add to this Everton team.
Ancelotti would agree that the more we see of this combative and forceful and skilful footballer, his shorts pulled high around his waist, the better.