Former Spurs winger Aaron Lennon struck completely against the run of play on 22 minutes and it took something special from Dele Alli to level it just before half-time.
Our dominance in that first period was backed up by the stats – 13 shots to Everton’s three, 65 per cent possession and both Harry Kane and Ben Davies struck the woodwork.
The second half started in the same pattern and it wasn’t until a late flurry that the hosts looked like they might turn the game around as first Muhamed Besic and then Ross Barkley drew saves from Hugo Lloris.
The wind and rain swirled around Goodison Park making conditions tricky underfoot with a saturated surface as we ran out in Merseyside unbeaten in our last six meetings with Everton.
Mauricio Pochettino reverted to a 4-2-3-1 system once more after a switch to three at the back against Watford. Christian Eriksen came in for the injured Mousa Dembele, while again both full-backs were rotated with Kyle Walker and Ben Davies preferred to Kieran Trippier and Danny Rose.
There was plenty of interest in the game from Belgium with Toby Alderweireld and Jan Vertonghen tasked with keeping quiet their international colleague Romelu Lukaku who, along with Harry Kane, is one of the Premier League’s most in-form strikers.
The rain eased from torrential to drizzle in time for kick-off and we almost broke the deadlock inside the opening five minutes when, after a good spell of possession, Walker drove into the box with purpose and fizzed a lethal looking ball across the face of goal that was desperately cleared with Alli lurking at the back post.
It was one-way traffic early on and we were a whisker away from the lead again five minutes later when Tom Carroll found Kane with a quick free-kick and he took aim on the angle from distance, but his shot rattled the foot of the post and skimmed back across the line before spinning behind for a goal kick.
The midfield was running smoothly with Carroll seeing plenty of the ball and Alli rooting out dangerous pockets of space in front of the Toffees’ back four.
Eriksen almost slipped Erik Lamela in soon afterwards but a fine saving block from Seamus Coleman denied him an opening goal.
It was arguably our best start to a game in terms of dominance this season, yet it remained goalless as the half ticked on and we were soon punished for failing to turn our possession into goals.
Lukaku peeled off Vertonghen and nodded a long diagonal ball into the path of Lennon, who produced a brilliant volley into the top corner from just inside the box.
The goal was very much against the run of play but kicked the home side into life as they started to play more inside our half.
We started to grasp control again, though, and once more rattled the woodwork with 30 minutes played. Eriksen picked out Davies with a clever corner routine and the defender took a touch before unleashing a thunderous shot from 25 yards that rattled the underside of the bar with Tim Howard beaten.
Again and again we knocked on the Everton door but still we trailed with the home side dropping deeper and making it difficult to breakthrough.
There was no room for a lapse in concentration though as, in Lukaku, the home side had a weapon capable of fashioning a chance from nothing and he looked a handful when involved.
It seemed like we would have to go in behind despite our dominance until a minute into first half stoppage time when we notched a deserved leveller. It was a goal made in West Bromwich as Alderweireld launched a raking long pass to Alli - much like he did a month ago at The Hawthorns - who gambled on the run and controlled brilliantly before volleying past Howard.
Neither side made any changes at the interval as the drizzle disappeared to leave us with a crisp January evening on Merseyside.
The hosts were quicker out the blocks but we almost went in front with five minutes of the second half played. Alli chested a ball from Davies into the path of Kane who looked certain to score but for a fine saving tackle from John Stones.
A minute later Eriksen drilled a low effort straight at Howard as we upped our game again but Everton looked more dangerous on the break after the restart.
Carroll was growing into the game with every pass and Alli looked intent on scoring another as the pair started to combine more frequently.
Lukaku was a centimetre away from a dangerous cross from Toffees substitute Gerard Deulofeu as, with 20 minutes remaining, it remained finely poised with both sides hopeful of winning the game.
It was the most open and even the game had been as the action swayed from one end to the other but as Everton searched for a winner, Lloris did well to tip over a dipping volley from substitute Besic before Ramiro Funes Mori glanced a header wide from the resulting corner.
It was end to end as both teams looked to land the knockout blow but there was only one more serious chance in the final few seconds as Barkley volleyed towards goal from the left angle, but Lloris was equal to it.
Everton (4-3-3): Howard; Coleman, Stones, Funes Mori, Baines; Barkley, Barry, Cleverley; Lennon (Deulofeu 59), Lukaku, Kone (Besic 59). Substitutes (not used): Robles, Galloway, Jagielka, Osman, Mirallas.
Spurs (4-2-3-1): Lloris (c), Walker, Alderweireld, Vertonghen, Davies; Carroll, Dier; Eriksen (Son 69), Alli (Chadli 84), Lamela (Onomah 89); Kane. Substitutes (not used): Vorm, Rose, Trippier, Bentaleb.
Goals: Everton – Lennon 22; Spurs - Alli 45+1.
Yellow cards: Everton – Coleman 35. Spurs – Lamela 68, Carroll 72.
Referee: Michael Oliver.