Micky van de Ven became the first defender to net two goals in a Premier League match for Tottenham since the Belgian defender in March 2013.
Match summary
The Spurs defender nods home twice in first half
Everton's Jake O'Brien has goal ruled after VAR review
Pape Matar Sarr nets final goal in dying moments
The defeat is Everton's first at their new ground
Micky van de Ven scored on two occasions as Spurs extended their unbeaten run in away games in the Premier League and handed Everton their first defeat at their new home.
The Toffees were undefeated in their five previous fixtures at their new ground at the dockside location, having left their historic home behind after 133 years at the conclusion of the previous campaign.
But Tottenham's leader the Dutchman struck on two occasions from corners in the opening 45 minutes to put their strong form at risk, and they were failed to turn it around before the midfielder - assisted by former Toffees attacker the Brazilian - nodded in a final goal to seal the victory.
Jake O'Brien thought he had levelled the tally five minutes after Van de Ven's first goal, glancing a powerful header past the Spurs keeper from an Everton corner.
The Portuguese striker, meanwhile, failed to convert a number of chances, while Vicario made a series of excellent saves to guarantee Tottenham went five matches on the road without defeat in the Premier League.
But the video assistant referee stepped in, advising Craig Pawson to check the score on the pitchside monitor and the referee ruled that Jack Grealish and the attacker were in an illegal position and had interfered with the Spurs keeper.
This victory elevates Thomas Frank's side into third place in the division, a handful of points off local foes and table-toppers Arsenal.
The Merseyside club stay in 14th position and are half a dozen points ahead of the drop zone.
Following a uninspiring weekday performance in the European competition against Monaco, where they managed a only two shots on target, Frank opted for the French striker up front instead of former Toffees player Richarlison.
The summer signing was unable to influence the game before being replaced for the substitute after 60 minutes on the pitch, but it emerged he proved unnecessary in the final outcome.
Centre-back the Dutchman converted into an attacking force to be feared.
Tottenham's first goal came from a well-worked set-piece play, with the midfielder heading the ball toward the face of goal for an free the defender to head home.
And Ven de Ven - filling in as skipper in Cristian Romero's unavailability - added his next goal in added time as he outjumped the England keeper to head home.
Van de Ven was just as impressive in defence, combining well with Kevin Danso as the pair handled superbly with the numerous deliveries thrown their way by Everton.
The club's new home had been a happy hunting ground, but it seems the freshness may have faded somewhat and supporters headed to the exits after the midfielder's third for Tottenham.
The end in Everton's winning streak at home won't be a massive worry for the manager, but their attacking woes are.
The forward and Thierno Barry - the Everton's two established strikers - had scored just a single goal between them before Sunday's match.
And they were unable to add to that miserable tally on a damp and dreary evening on in Liverpool.
The striker's performance was epitomized when he opted to bicycle-kick a clear chance from near the goal, even with having room and time to reposition for a more conventional attempt that would have challenged the keeper.
The substitute, too, failed to influence the game after replacing the starter in the second half.
It was not for a lack of opportunity. The majority of Everton's offensive moves came through Grealish and the forward, who relentlessly whipped dangerous crosses into the Spurs penalty area.
The Toffees: Moyes' men head to the Stadium of Light in the top flight on early November at 8 PM.
Tottenham: The London club go to St James' Park on late October for their fourth-round League Cup tie against the Magpies.
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