Micky van de Ven became the first backline player to score two goals in a English top-flight game for Tottenham since the Belgian defender in spring 2013.
Key points
The Spurs defender heads into the net two times in opening period
The home side's Jake O'Brien has equalizer ruled after video check
Senegalese midfielder nets third in dying moments
The defeat is Everton's first at Hill Dickinson Stadium
Micky van de Ven found the net on two occasions as Spurs extended their unbeaten run in away games in the Premier League and handed Everton their initial loss at their new home.
Everton were unbeaten in their last five matches at their recently opened stadium at the dockside location, having left Goodison Park behind after 133 years at the conclusion of the previous campaign.
But Tottenham's leader the Dutchman struck twice from corners in the opening 45 minutes to place their strong form at jeopardy, and they were failed to mount a comeback before the midfielder - set up by former Toffees attacker Richarlison - nodded in a third to seal the victory.
The Everton defender thought he had levelled the tally a short while after the defender's opener, glancing a powerful header past Guglielmo Vicario from an set-piece.
Beto, meanwhile, failed to convert a handful of chances, while Vicario made a series of fine stops to guarantee Tottenham went five away games unbeaten in the Premier League.
But the VAR intervened, advising the match official to check the goal on the screen and the official decided that Jack Grealish and the attacker were in an illegal position and had impeded with the Tottenham goalkeeper.
The win moves Thomas Frank's team into third place in the division, a handful of points off north London rivals and table-toppers Arsenal.
The Merseyside club remain 14th and are half a dozen points clear of the drop zone.
Following a drab weekday performance in the Champions League against the French side, where they managed a only two shots on goal, the manager chose Randal Kolo Muani up front instead of former Toffees player Richarlison.
The summer signing was failed to make much of an impact before being replaced for Richarlison after an hour on the field, but it turns out he wasn't needed in the final outcome.
Centre-back Van de Ven converted into an offensive threat to be reckoned with.
Tottenham's first goal came from a cleverly executed corner routine, with Rodrigo Bentancur directing the ball toward the six-yard box for an free Van de Ven to head home.
And Ven de Ven - filling in as skipper in Cristian Romero's unavailability - added his second in added time as he rose above Jordan Pickford to head home.
The Dutchman was just as outstanding in defence, combining well with his defensive colleague as the pair handled superbly with the numerous deliveries thrown their way by the opposition.
The club's new home had been a happy hunting ground, but it appears the novelty may have worn off slightly and fans headed to the gates after the midfielder's third for Spurs.
The break in the team's winning streak at home will not be a major concern for the manager, but their attacking woes will be.
The forward and the other striker - the Everton's two established strikers - had scored just one league goal combined before the weekend's outing.
And they were failed to add to that poor tally on a damp and dreary evening on Merseyside.
Beto's performance was summed up when he chose to bicycle-kick a clear chance from close range, even with having room and time to readjust for a simpler shot that would have challenged the keeper.
Barry, as well, was unable to make an impact after replacing the starter in the second half.
It was not for a shortage of opportunity. Most of Everton's offensive moves came through Grealish and the forward, who relentlessly delivered threatening crosses into the Spurs box.
Everton: Moyes' men head to the Stadium of Light in the top flight on Monday, 3 November at 8 PM.
Tottenham: Spurs travel to St James' Park on Wednesday, 29 October for their next round League Cup match against Newcastle.
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