Micky van de Ven emerged as the first backline player to net twice in a Premier League match for Tottenham since the Belgian defender in spring 2013.
At a glance
The Spurs defender nods home two times in opening period
Everton's Jake O'Brien has equalizer ruled after video check
Pape Matar Sarr nets final goal in dying moments
This loss is the Merseyside club's first at their new ground
The Tottenham defender scored twice as Tottenham extended their winning streak in away games in the Premier League and delivered Everton their first defeat at their new home.
The Toffees were undefeated in their five previous fixtures at their recently opened stadium at the dockside location, having left Goodison Park behind after more than a century at the conclusion of the previous campaign.
But Tottenham's leader Van de Ven pounced on two occasions from set-pieces in the first half to place their strong form at jeopardy, and they were failed to turn it around before the midfielder - assisted by ex-Everton forward Richarlison - headed home a third to put the game to bed.
Jake O'Brien believed he had equalized the scores a short while after the defender's first goal, glancing a powerful header past Guglielmo Vicario from an set-piece.
Beto, meanwhile, missed a handful of opportunities, while the Italian goalkeeper made a series of fine saves to guarantee the London club went five matches on the road without defeat in the English top flight.
But the video assistant referee intervened, instructing the match official to check the goal on the screen and the official ruled that the winger and Iliman Ndiaye were in an illegal position and had interfered with the Tottenham goalkeeper.
The win moves Thomas Frank's side up to third in the Premier League, five points off north London rivals and table-toppers Arsenal.
Everton remain in 14th position and are six points ahead of the relegation zone.
Following a uninspiring midweek performance in the Champions League against Monaco, where they mustered a mere two shots on target, the manager chose Randal Kolo Muani in attack instead of ex-Everton forward Richarlison.
The new arrival was unable to influence the game before being replaced for Richarlison after 60 minutes on the field, but it emerged he wasn't needed in the final outcome.
Defender Van de Ven transformed into an attacking force to be reckoned with.
Spurs' first goal came from a cleverly executed set-piece play, with the midfielder heading the ball back across the face of goal for an free Van de Ven to score.
And the captain - stepping up as skipper in Cristian Romero's unavailability - added his second in first-half stoppage time as he rose above the England keeper to head home.
Van de Ven was just as outstanding in defence, partnering effectively with his defensive colleague as the duo handled superbly with the 34 crosses thrown their way by Everton.
The club's new home had been a happy hunting ground, but it appears the freshness may have faded slightly and fans flocked to the exits after Sarr's third for Spurs.
The end in the team's unbeaten home run won't be a massive worry for David Moyes, but their striker situation are.
Beto and Thierno Barry - the Toffees' two recognised centre-forwards - had netted just one league goal between them before the weekend's match.
And they were failed to contribute to that miserable tally on a wet and gloomy evening on Merseyside.
The striker's performance was epitomized when he chose to overhead kick a golden opportunity from close range, despite having space and opportunity to readjust for a simpler attempt that would have challenged the keeper.
Barry, too, failed to influence the game after replacing Beto in the 65th minute.
It wasn't for a lack of chance. The majority of the home side's offensive moves came through the winger and the forward, who tirelessly delivered threatening crosses into the Tottenham box.
The Toffees: Moyes' men travel to Sunderland in the Premier League on Monday, 3 November at 20:00 GMT.
Tottenham: The London club go to Newcastle's ground on Wednesday, 29 October for their fourth-round Carabao Cup tie against Newcastle.
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