The Dutch defender emerged as the first defender to net two goals in a Premier League match for Tottenham since Jan Vertonghen in March 2013.
Match summary
The Spurs defender nods into the net two times in first half
The home side's defender has goal ruled after VAR review
Senegalese midfielder scores final goal in 89th minute
The defeat is the Merseyside club's initial at Hill Dickinson Stadium
Micky van de Ven scored twice as Tottenham prolonged their winning streak in away games in the Premier League and handed the Toffees their first defeat at their new home.
Everton were unbeaten in their five previous fixtures at their recently opened stadium at the dockside location, having departed Goodison Park behind after 133 years at the conclusion of the previous campaign.
But Tottenham's leader Van de Ven pounced twice from corners in the first half to place their strong form at risk, and they were unable to turn it around before Pape Matar Sarr - set up by ex-Everton forward the Brazilian - nodded in a third to put the game to bed.
Jake O'Brien thought he had equalized the tally five minutes after Van de Ven's opener, directing a powerful header past Guglielmo Vicario from an Everton corner.
Beto, in the meantime, failed to convert a handful of opportunities, while Vicario made a succession of excellent saves to ensure Tottenham went five matches on the road unbeaten in the Premier League.
But the video assistant referee intervened, advising the match official to review the score on the screen and the official decided that the winger and Iliman Ndiaye were in offside positions and had interfered with the Spurs keeper.
The win moves the manager's team up to third in the Premier League, a handful of points off local foes and table-toppers Arsenal.
The Merseyside club stay in 14th position and sit half a dozen points ahead of the relegation zone.
After a uninspiring weekday display in the Champions League against the French side, where they mustered a only two shots on target, the manager chose the French striker in attack instead of ex-Everton forward the Brazilian.
The summer signing was unable to influence the game before being replaced for Richarlison after an hour on the field, but it turns out he wasn't needed in the end.
Centre-back the Dutchman converted into an offensive threat to be reckoned with.
Tottenham's opening strike came from a well-worked corner routine, with the midfielder heading the ball back across the face of goal for an free Van de Ven to head home.
And the captain - filling in as captain in Cristian Romero's absence - added his second in added time as he rose above Jordan Pickford to head home.
The Dutchman was equally impressive in defence, partnering effectively with Kevin Danso as the duo handled superbly with the numerous deliveries thrown their way by the opposition.
Everton's new home had been a fortress, but it appears the novelty may have worn off somewhat and supporters headed to the exits after the midfielder's third for Spurs.
The break in Everton's winning streak at home will not be a major worry for David Moyes, but their striker situation will be.
The forward and the other striker - the Toffees' pair of established strikers - had scored just a single goal between them before Sunday's match.
And they were failed to add to that miserable tally on a damp and dreary night on Merseyside.
The striker's form was epitomized when he opted to bicycle-kick a clear chance from close range, despite having room and opportunity to readjust for a more conventional shot that would have challenged the keeper.
Barry, as well, failed to influence the game after replacing Beto in the 65th minute.
It wasn't for a shortage of opportunity. Most of the home side's offensive moves came through the winger and the forward, who relentlessly delivered threatening balls into the Tottenham penalty area.
The Toffees: Moyes' men travel to the Stadium of Light in the top flight on early November at 8 PM.
Spurs: The London club travel to Newcastle's ground on Wednesday, 29 October for their fourth-round League Cup match against the Magpies.