Micky van de Ven became the initial backline player to score twice in a Premier League game for Spurs since the Belgian defender in March 2013.
Match summary
The Spurs defender heads into the net twice in opening period
Everton's Jake O'Brien has equalizer ruled after video check
Senegalese midfielder scores third in dying moments
The defeat is Everton's initial at their new ground
The Tottenham defender found the net twice as Spurs extended their unbeaten run in away games in the Premier League and delivered Everton their first defeat at Hill Dickinson Stadium.
Everton were undefeated in their five previous fixtures at their new ground at the dockside location, having left Goodison Park behind after more than a century at the conclusion of last season.
But Tottenham's leader the Dutchman struck twice from set-pieces in the opening 45 minutes to put their strong form at risk, and they were failed to turn it around before Pape Matar Sarr - assisted by ex-Everton forward Richarlison - nodded in a third to seal the victory.
Jake O'Brien thought he had levelled the tally a short while after Van de Ven's opener, glancing a powerful header past Guglielmo Vicario from an set-piece.
Beto, meanwhile, missed a handful of opportunities, while Vicario made a succession of excellent saves to ensure Tottenham went five matches on the road without defeat in the English top flight.
But the video assistant referee stepped in, instructing the match official to review the goal on the screen and the referee decided that the winger and the attacker were in an illegal position and had interfered with the Spurs keeper.
The win moves Thomas Frank's side into third place in the division, five points off north London rivals and leaders Arsenal.
Everton stay in 14th position and sit six points ahead of the relegation zone.
Following a uninspiring weekday performance in the European competition against the French side, where they mustered a only attempts on target, the manager chose the French striker in attack instead of former Toffees player Richarlison.
The new arrival was unable to influence the game before being replaced for Richarlison after 60 minutes on the pitch, but it emerged he proved unnecessary in the end.
Centre-back the Dutchman converted into an attacking force to be feared.
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 unmarked Van de Ven to score.
And the captain - filling in as skipper in Cristian Romero's unavailability - added his second in first-half stoppage time as he rose above Jordan Pickford to nod in.
The Dutchman was equally outstanding in defensive duties, partnering effectively with Kevin Danso as the pair dealt superbly with the numerous deliveries thrown their way by Everton.
Everton's recent stadium had been a happy hunting ground, but it appears the freshness may have faded slightly and supporters headed to the exits after the midfielder's final goal for Tottenham.
The break in Everton's unbeaten home run will not be a massive concern for David Moyes, but their striker situation are.
The forward and Thierno Barry - the Everton's pair of established centre-forwards - had scored just one league goal between them before Sunday's match.
And they were failed to add to that miserable record on a damp and dreary evening on in Liverpool.
Beto's performance was summed up when he opted to overhead kick a clear chance from close range, even with having space and opportunity to readjust for a simpler shot that would have challenged the keeper.
Barry, as well, was unable to make an impact after replacing Beto in the second half.
It was not for a lack of chance. The majority of Everton's offensive moves came through the winger and the forward, who tirelessly delivered threatening balls into the Tottenham penalty area.
Everton: The team travel to the Stadium of Light in the Premier League on Monday, 3 November at 20:00 GMT.
Spurs: The London club travel to Newcastle's ground on late October for their fourth-round Carabao Cup tie against Newcastle.
A seasoned journalist with a background in political science, specializing in breaking down complex news into accessible briefs.