The Dutch defender emerged as the first defender to net twice in a English top-flight game for Tottenham since Jan Vertonghen in March 2013.
At a glance
The Spurs defender heads home twice in first half
The home side's Jake O'Brien has equalizer ruled after video check
Senegalese midfielder scores third in dying moments
The defeat is Everton's initial at their new ground
The Tottenham defender scored twice as Spurs extended their winning streak in away games in the Premier League and delivered the Toffees their first defeat at Hill Dickinson Stadium.
The Toffees were undefeated in their five previous fixtures at their new ground at Bramley-Moore Dock, having departed their historic home behind after more than a century at the conclusion of last season.
But Tottenham's leader the Dutchman struck on two occasions from corners in the first half to put their strong form at risk, and they were unable to turn it around before the midfielder - assisted by former Toffees attacker Richarlison - headed home a final goal to put the game to bed.
The Everton defender believed he had levelled the scores a short while after the defender's first goal, glancing a powerful header past the Spurs keeper from an Everton corner.
Beto, meanwhile, failed to convert a handful of opportunities, while the Italian goalkeeper made a succession of excellent stops to ensure the London club went five matches on the road without defeat in the English top flight.
But the video assistant referee stepped in, advising the match official to review the goal on the pitchside monitor and the referee decided that Jack Grealish and the attacker were in offside positions and had impeded with the Spurs keeper.
The win elevates Thomas Frank's team into third place in the division, five points off local foes and table-toppers Arsenal.
Everton stay 14th and are six points clear of the drop zone.
Following a drab weekday performance in the Champions League against Monaco, where they managed a only two shots on goal, Frank opted for Randal Kolo Muani up front instead of ex-Everton forward Richarlison.
The new arrival was unable to make much of an impact before being replaced for the substitute after an hour on the pitch, but it emerged he wasn't needed in the final outcome.
Centre-back Van de Ven transformed into an attacking force to be reckoned with.
Tottenham's first goal came from a well-worked set-piece play, with the midfielder heading the ball back across the face of goal for an free Van de Ven to head home.
And Ven de Ven - stepping up as captain in Cristian Romero's unavailability - notched his second in added time as he outjumped Jordan Pickford to head home.
Van de Ven was equally outstanding in defensive duties, combining effectively with Kevin Danso as the duo dealt superbly with the numerous deliveries sent their way by Everton.
The club's new home had been a happy hunting ground, but it appears the freshness may have faded slightly and supporters headed to the exits after Sarr's final goal for Tottenham.
The break in the team's winning streak at home will not be a major worry for the manager, but their attacking woes will be.
The forward and Thierno Barry - the Toffees' pair of established centre-forwards - had scored only a single goal combined before Sunday's outing.
And they were failed to add to that poor tally on a wet and gloomy evening on Merseyside.
The striker's performance was epitomized when he chose to overhead kick a golden opportunity from near the goal, even with having space and opportunity to reposition for a more conventional shot that would have challenged the keeper.
Barry, too, was unable to influence the game after coming on for Beto in the second half.
It wasn't for a shortage of chance. The majority of the home side's offensive moves came through Grealish and the forward, who relentlessly whipped threatening balls into the Spurs penalty area.
The Toffees: Moyes' men travel to the Stadium of Light in the top flight on early November at 8 PM.
Spurs: Spurs travel to Newcastle's ground on late October for their next round League Cup match against Newcastle.