Micky van de Ven became the first backline player to net twice in a English top-flight game for Spurs since the Belgian defender in March 2013.
Match summary
Micky Van de Ven nods into the net two times in first half
Everton's Jake O'Brien has equalizer ruled after video check
Pape Matar Sarr scores third in dying moments
The defeat is Everton's initial at their new ground
Micky van de Ven scored on two occasions as Spurs extended their winning streak in away games in the top division and handed Everton their initial loss at their new home.
Everton were undefeated in their five previous fixtures at their new ground at Bramley-Moore Dock, having departed their historic home behind after 133 years at the end of the previous campaign.
But Spurs captain the Dutchman pounced twice from set-pieces in the opening 45 minutes to place their strong form at risk, and they were failed to mount a comeback before Pape Matar Sarr - assisted by former Toffees attacker the Brazilian - headed home a final goal to seal the victory.
Jake O'Brien thought he had equalized the scores five minutes after Van de Ven's opener, directing a powerful header past the Spurs keeper from an set-piece.
The Portuguese striker, in the meantime, missed a number of chances, while the Italian goalkeeper made a succession of fine saves to ensure Tottenham went five matches on the road without defeat in the English top flight.
But the VAR intervened, instructing the match official to review the goal on the screen and the official decided that the winger and the attacker were in an illegal position and had impeded with the Spurs keeper.
This victory moves the manager's side into third place in the Premier League, a handful of points off north London rivals and leaders the Gunners.
Everton stay in 14th position and sit half a dozen points ahead of the relegation zone.
Following a uninspiring midweek display in the Champions League against the French side, where they managed a only attempts on target, the manager opted for the French striker in attack instead of former Toffees player the Brazilian.
The new arrival was failed to influence the game before being substituted for Richarlison after 60 minutes on the pitch, but it turns out he wasn't needed in the final outcome.
Centre-back Van de Ven converted into an offensive threat to be feared.
Spurs' first goal came from a well-worked set-piece play, with Rodrigo Bentancur directing the ball back across the six-yard box for an free Van de Ven to head home.
And the captain - filling in as captain in Cristian Romero's absence - notched his next goal in first-half stoppage time as he rose above the England keeper to head home.
The Dutchman was just as outstanding in defence, partnering well with Kevin Danso as the pair dealt excellently with the numerous deliveries sent their way by the opposition.
The club's new home had been a fortress, but it seems the novelty may have worn off slightly and fans flocked to the gates after the midfielder's third for Spurs.
The break in the team's unbeaten home run will not be a massive concern for the manager, but their striker situation are.
The forward and Thierno Barry - the Toffees' two established strikers - had netted just one league goal combined before the weekend's match.
And they were unable to contribute to that poor tally on a wet and gloomy night on Merseyside.
Beto's performance was summed up when he chose to bicycle-kick a clear chance from close range, despite having room and opportunity to reposition for a more conventional attempt that would have challenged the keeper.
The substitute, too, was unable to influence the game after replacing Beto in the 65th minute.
It wasn't for a shortage of chance. Most of Everton's offensive moves came through Grealish and Ndiaye, who tirelessly whipped dangerous crosses into the Spurs box.
The Toffees: The team head to the Stadium of Light in the top flight on early November at 20:00 GMT.
Tottenham: The London club go to St James' Park on late October for their next round League Cup match against Newcastle.
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