The Dutch defender emerged as the first defender to score twice in a Premier League game for Spurs since the Belgian defender in March 2013.
Key points
The Spurs defender nods into the net twice in first half
The home side's defender has equalizer disallowed after VAR review
Senegalese midfielder scores final goal in 89th minute
This loss is Everton's first at Hill Dickinson Stadium
The Tottenham defender scored twice as Spurs prolonged their unbeaten run in away games in the Premier League and delivered Everton their first defeat at their new home.
Everton were unbeaten in their last five matches at their recently opened stadium at the dockside location, having departed Goodison Park behind after 133 years at the conclusion of the previous campaign.
But Spurs captain the Dutchman struck on two occasions from set-pieces in the first half to put their strong form at risk, and they were unable to turn it around before the midfielder - set up by ex-Everton forward Richarlison - headed home a third to seal the victory.
Jake O'Brien believed he had equalized the tally a short while after Van de Ven's opener, directing a powerful header past the Spurs keeper from an set-piece.
Beto, meanwhile, missed a number of opportunities, while Vicario made a series of fine stops to guarantee the London club went five away games unbeaten in the English top flight.
But the VAR stepped in, instructing Craig Pawson to check the score on the pitchside monitor and the official decided that the winger and Iliman Ndiaye were in an illegal position and had impeded with the Spurs keeper.
This victory moves Thomas Frank's team up to third in the Premier League, five points off north London rivals and table-toppers Arsenal.
The Merseyside club stay 14th and are half a dozen points ahead of the relegation zone.
Following a uninspiring weekday display in the Champions League against the French side, where they managed a mere attempts on target, the manager opted for Randal Kolo Muani up front instead of former Toffees player Richarlison.
The new arrival was unable to influence the game before being replaced for the substitute after an hour on the pitch, but it turns out he wasn't needed in the end.
Centre-back the Dutchman transformed into an attacking force to be reckoned with.
Tottenham's opening strike came from a cleverly executed set-piece play, with Rodrigo Bentancur directing the ball toward the six-yard box for an free the defender to head home.
And Ven de Ven - filling in as captain in the Argentine's absence - added his next goal in added time as he rose above Jordan Pickford to nod in.
The Dutchman was just as outstanding in defence, combining effectively with Kevin Danso as the duo dealt excellently with the 34 crosses sent their way by the opposition.
Everton's new home had been a fortress, but it seems the freshness may have faded slightly and supporters flocked to the gates after Sarr's third for Spurs.
The break in Everton's unbeaten home run won't be a major worry for the manager, but their attacking woes will be.
Beto and the other striker - the Toffees' two recognised centre-forwards - had netted only one league goal between them before Sunday's match.
And they were unable to contribute to that poor record on a damp and dreary evening on in Liverpool.
The striker's form was summed up when he opted to overhead kick a clear chance from close range, despite having space and opportunity to readjust for a simpler attempt that would have challenged the keeper.
Barry, too, was unable to make an impact after coming on for Beto in the second half.
It wasn't for a lack of opportunity. The majority of Everton's offensive moves came through the winger and Ndiaye, who tirelessly whipped threatening balls into the Spurs box.
Everton: The team head to Sunderland in the Premier League on Monday, 3 November at 20:00 GMT.
Tottenham: The London club travel to Newcastle's ground on Wednesday, 29 October for their next round Carabao Cup tie against the Magpies.