The Dutch defender emerged as the initial defender to score twice in a English top-flight match for Spurs since Jan Vertonghen in March 2013.
Match summary
The Spurs defender heads into the net two times in opening period
Everton's defender has goal ruled after video check
Senegalese midfielder scores third in 89th minute
This loss is Everton's first at Hill Dickinson Stadium
The Tottenham defender scored twice as Spurs extended their unbeaten run on the road in the top division and handed the Toffees their first defeat at Hill Dickinson Stadium.
The Toffees were undefeated in their five previous fixtures at their recently opened stadium at Bramley-Moore Dock, having departed Goodison Park behind after more than a century at the conclusion of the previous campaign.
But Tottenham's leader the Dutchman pounced twice from corners in the opening 45 minutes to place their strong form at jeopardy, and they were failed to turn it around before Pape Matar Sarr - assisted by former Toffees attacker the Brazilian - nodded in a third to put the game to bed.
The Everton defender thought he had levelled the scores five minutes after the defender's first goal, directing a bullet header past the Spurs keeper from an Everton corner.
The Portuguese striker, in the meantime, failed to convert a handful of opportunities, while Vicario made a series of excellent stops to guarantee Tottenham went five away games without defeat in the Premier League.
But the VAR stepped in, instructing Craig Pawson to check the score on the pitchside monitor and the official ruled that the winger and Iliman Ndiaye were in offside positions and had interfered with the Tottenham goalkeeper.
This victory elevates the manager's side into third place in the division, a handful of points off local foes and table-toppers the Gunners.
The Merseyside club stay in 14th position and are six points clear of the relegation zone.
After a uninspiring midweek display in the Champions League against Monaco, where they mustered a only two shots on target, Frank opted for the French striker up front instead of ex-Everton forward the Brazilian.
The new arrival 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 end.
Centre-back the Dutchman transformed into an attacking force to be reckoned with.
Tottenham's first goal came from a well-worked corner routine, with Rodrigo Bentancur directing the ball back across the six-yard box for an free Van de Ven to score.
And Ven de Ven - stepping up as skipper in the Argentine's unavailability - notched his next goal in added time as he outjumped Jordan Pickford to head home.
The Dutchman was equally impressive in defence, partnering effectively with Kevin Danso as the pair handled superbly with the 34 crosses sent their way by Everton.
Everton's new home had been a fortress, but it seems the novelty may have faded slightly and fans flocked to the gates after Sarr's final goal for Spurs.
The break in the team's unbeaten home run will not be a massive worry for David Moyes, but their attacking woes are.
The forward and the other striker - the Toffees' two established centre-forwards - had scored just a single goal between them before the weekend's outing.
And they were failed to contribute to that poor record on a damp and dreary 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 space and time to reposition for a simpler attempt that would have challenged the keeper.
Barry, as well, failed to make an impact after coming on for the starter in the second half.
It wasn't for a shortage of chance. Most of Everton's attacks came through the winger and Ndiaye, who relentlessly whipped threatening crosses into the Spurs box.
Everton: Moyes' men travel to the Stadium of Light in the top flight on Monday, 3 November at 20:00 GMT.
Spurs: The London club go to Newcastle's ground on Wednesday, 29 October for their next round Carabao Cup tie against Newcastle.
Liam Sterling is a seasoned betting analyst with over a decade of experience in the online gambling industry.