Micky van de Ven became the first backline player to net two goals in a Premier League game for Tottenham since the Belgian defender in spring 2013.
Match summary
The Spurs defender nods home two times in opening period
Everton's defender has equalizer disallowed after VAR review
Senegalese midfielder scores final goal in 89th minute
The defeat is the Merseyside club's first at their new ground
The Tottenham defender found the net on two occasions as Spurs extended their winning streak in away games in the Premier League and handed Everton their first defeat at their new home.
The Toffees were undefeated in their last five matches at their recently opened stadium at the dockside location, having departed their historic home behind after more than a century at the end of last season.
But Spurs captain the Dutchman struck twice from set-pieces in the first half to place their strong form at risk, and they were unable to turn it around before Pape Matar Sarr - assisted by former Toffees attacker Richarlison - headed home a final goal to seal the victory.
Jake O'Brien thought he had equalized the scores a short while after Van de Ven's first goal, directing a powerful header past the Spurs keeper from an set-piece.
Beto, meanwhile, missed a handful of chances, while Vicario made a series of excellent stops to guarantee the London club went five matches on the road without defeat in the English top flight.
But the video assistant referee stepped in, instructing Craig Pawson to check the score on the pitchside monitor and the referee decided that the winger and the attacker were in an illegal position and had interfered with the Tottenham goalkeeper.
The win moves Thomas Frank's side into third place in the division, five points off local foes and table-toppers the Gunners.
The Merseyside club remain 14th and sit half a dozen points ahead of the relegation zone.
After a drab midweek performance in the European competition against the French side, where they mustered a mere attempts on goal, Frank chose the French striker in attack instead of former Toffees player Richarlison.
The new arrival was unable to make much of an impact before being replaced for the substitute after 60 minutes on the pitch, but it turns out he wasn't needed in the end.
Defender Van de Ven converted into an offensive threat to be reckoned with.
Tottenham's opening strike came from a cleverly executed set-piece play, with Rodrigo Bentancur heading the ball toward the six-yard box for an free the defender to head home.
And Ven de Ven - stepping up as captain in the Argentine's absence - notched his next goal in first-half stoppage time as he rose above Jordan Pickford to head home.
Van de Ven was just as outstanding in defence, combining well with Kevin Danso as the duo dealt excellently with the numerous deliveries thrown their way by the opposition.
Everton's recent stadium had been a fortress, but it seems the freshness may have worn off somewhat and fans headed to the exits after the midfielder's third for Spurs.
The break in Everton's unbeaten home run will not be a massive worry for the manager, but their striker situation are.
Beto and Thierno Barry - the Everton's two established strikers - had scored only one league goal combined before the weekend's outing.
And they were failed to add to that miserable tally on a wet and gloomy night on Merseyside.
The striker's form was epitomized when he chose to bicycle-kick a golden opportunity from close range, even with having room and opportunity to readjust for a more conventional attempt that would have tested the keeper.
The substitute, as well, failed to make an impact after replacing the starter in the 65th minute.
It wasn't for a shortage of chance. The majority of Everton's offensive moves came through the winger and Ndiaye, who tirelessly delivered dangerous balls into the Tottenham penalty area.
The Toffees: The team travel to the Stadium of Light in the top flight on early November at 20:00 GMT.
Spurs: The London club travel to Newcastle's ground on Wednesday, 29 October for their fourth-round Carabao Cup match against Newcastle.
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