The Dutch defender emerged as the initial defender to score twice in a Premier League match for Spurs since Jan Vertonghen in spring 2013.
Key points
The Spurs defender heads into the net two times in opening period
Everton's Jake O'Brien has equalizer disallowed after VAR review
Pape Matar Sarr scores third in dying moments
This loss is Everton's first at Hill Dickinson Stadium
Micky van de Ven scored twice as Tottenham extended their winning streak in away games in the Premier League and handed Everton their initial loss at their new home.
Everton were unbeaten in their five previous fixtures at their recently opened stadium at the dockside location, having departed Goodison Park behind after 133 years at the end of the previous campaign.
But Tottenham's leader the Dutchman pounced twice from set-pieces in the opening 45 minutes to place their impressive record at jeopardy, and they were unable to mount a comeback before Pape Matar Sarr - set up by ex-Everton forward Richarlison - nodded in a third to seal the victory.
The Everton defender believed he had equalized the tally a short while after the defender's opener, glancing a bullet header past the Spurs keeper from an Everton corner.
The Portuguese striker, in the meantime, failed to convert a handful of chances, while Vicario made a series of fine stops to ensure the London club went five away games unbeaten in the English top flight.
But the VAR intervened, advising the match official to review the goal on the screen and the official decided that Jack Grealish and Iliman Ndiaye were in an illegal position and had impeded with the Tottenham goalkeeper.
This victory elevates the manager's team into third place in the Premier League, five points off local foes and table-toppers Arsenal.
The Merseyside club stay in 14th position and sit half a dozen points clear of the relegation zone.
After a drab weekday performance in the Champions League against Monaco, where they mustered a mere two shots on goal, Frank chose the French striker in attack instead of ex-Everton forward the Brazilian.
The summer signing was failed to make much of an impact before being substituted for the substitute after an hour on the field, but it turns out he proved unnecessary in the final outcome.
Defender Van de Ven transformed into an attacking force to be feared.
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 the defender to score.
And the captain - filling in as captain in the Argentine's unavailability - notched his next goal in added time as he rose above the England keeper to nod in.
The Dutchman was equally impressive in defensive duties, partnering well with his defensive colleague as the pair handled excellently with the numerous deliveries thrown their way by Everton.
Everton's new home had been a fortress, but it seems the freshness may have faded somewhat and supporters headed to the gates after the midfielder's third for Spurs.
The end in the team's unbeaten home run won't be a major concern for David Moyes, but their striker situation will be.
Beto and Thierno Barry - the Everton's two established strikers - had scored only one league goal combined before Sunday's match.
And they were failed to add to that miserable tally on a damp and dreary night on in Liverpool.
The striker's form was summed up when he chose to bicycle-kick a clear chance from close range, even with having room and opportunity to reposition for a more conventional shot that would have challenged the keeper.
The substitute, as well, failed to make an impact after replacing the starter in the second half.
It wasn't for a shortage of chance. Most of Everton's offensive moves came through Grealish and the forward, who tirelessly delivered threatening balls into the Spurs box.
Everton: The team head to the Stadium of Light in the top flight on Monday, 3 November at 20:00 GMT.
Spurs: Spurs travel to Newcastle's ground on Wednesday, 29 October for their fourth-round League Cup tie against the Magpies.
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