Micky van de Ven became the initial defender to net two goals in a Premier League match for Spurs since the Belgian defender in spring 2013.
At a glance
Micky Van de Ven heads home twice in first half
The home side's Jake O'Brien has goal ruled after video check
Pape Matar Sarr scores third in 89th minute
The defeat is Everton's first at Hill Dickinson Stadium
The Tottenham defender found the net on two occasions as Spurs prolonged their unbeaten run in away games in the Premier League and delivered the Toffees their first defeat at Hill Dickinson Stadium.
The Toffees were undefeated in their last five matches at their new ground at the dockside location, having departed Goodison Park behind after 133 years at the conclusion of the previous campaign.
But Tottenham's leader the Dutchman pounced on two occasions from corners in the opening 45 minutes to put their strong form at risk, and they were failed to mount a comeback before the midfielder - assisted by ex-Everton forward Richarlison - headed home a third to put the game to bed.
The Everton defender believed he had levelled the tally five minutes after the defender's opener, directing a powerful header past the Spurs keeper from an set-piece.
The Portuguese striker, meanwhile, failed to convert a handful of chances, while Vicario made a succession of fine saves to ensure the London club went five away games without defeat in the Premier League.
But the VAR stepped in, advising Craig Pawson to check the goal on the pitchside monitor and the official ruled that the winger and the attacker were in offside positions and had interfered with the Tottenham goalkeeper.
This victory elevates the manager's team into third place in the Premier League, a handful of points off local foes and leaders Arsenal.
The Merseyside club remain in 14th position and are six points ahead of the drop zone.
Following a drab weekday performance in the Champions League against the French side, where they mustered a mere two shots on target, the manager chose the French striker up front instead of former Toffees player the Brazilian.
The summer signing was failed to influence the game before being substituted for Richarlison after an hour on the pitch, but it emerged 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 cleverly executed corner routine, with Rodrigo Bentancur heading the ball back across the face of goal for an free Van de Ven to score.
And the captain - filling in as skipper in the Argentine's absence - notched his next goal in added time as he rose above Jordan Pickford to nod in.
The Dutchman was just as outstanding in defence, partnering effectively with Kevin Danso as the pair handled superbly with the numerous deliveries sent their way by Everton.
Everton's recent stadium had been a fortress, but it seems the novelty may have faded slightly and fans headed to the gates after Sarr's third for Spurs.
The end in Everton's unbeaten home run won't be a massive concern for David Moyes, but their striker situation will be.
The forward and the other striker - the Everton's pair of established strikers - had scored just a single goal combined before Sunday's match.
And they were failed to add to that miserable tally on a wet and gloomy evening on in Liverpool.
Beto's performance was summed up when he chose to overhead kick a golden opportunity from near the goal, even with having room and time to reposition for a more conventional attempt that would have challenged Vicario.
Barry, too, failed to make an impact after replacing Beto in the 65th minute.
It wasn't for a shortage of chance. The majority of the home side's attacks came through the winger and Ndiaye, who tirelessly whipped threatening crosses into the Spurs penalty area.
Everton: Moyes' men travel to Sunderland in the top flight on Monday, 3 November at 8 PM.
Spurs: The London club go to St James' Park on Wednesday, 29 October for their next round Carabao Cup tie against the Magpies.
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