The Dutch defender became the initial backline player to score two goals in a Premier League match for Spurs since the Belgian defender in spring 2013.
Match summary
The Spurs defender nods into the net two times in first half
Everton's Jake O'Brien has goal ruled after VAR review
Pape Matar Sarr scores final goal in dying moments
This loss is the Merseyside club's initial at their new ground
The Tottenham defender scored on two occasions as Spurs prolonged their winning streak in away games in the top division and delivered the Toffees their first defeat at Hill Dickinson Stadium.
The Toffees were unbeaten in their five previous fixtures at their recently opened stadium at the dockside location, having departed their historic home behind after more than a century at the end of last season.
But Tottenham's leader Van de Ven pounced on two occasions from set-pieces in the opening 45 minutes to put their impressive record at jeopardy, and they were failed to mount a comeback before Pape Matar Sarr - assisted by former Toffees attacker Richarlison - headed home a third to seal the victory.
Jake O'Brien believed he had equalized the scores a short while after Van de Ven's opener, glancing a powerful header past the Spurs keeper from an Everton corner.
Beto, in the meantime, failed to convert a number of opportunities, while Vicario made a succession of fine saves to guarantee Tottenham went five matches on the road unbeaten in the English top flight.
But the video assistant referee intervened, advising Craig Pawson to check the goal on the screen and the referee ruled that Jack Grealish and the attacker were in an illegal position and had impeded with the Spurs keeper.
This victory moves the manager's team up to third in the division, a handful of points off local foes and leaders Arsenal.
The Merseyside club remain in 14th position and sit half a dozen points clear of the drop zone.
Following a drab midweek display in the European competition against Monaco, where they managed a mere two shots on target, Frank chose the French striker in attack instead of ex-Everton forward the Brazilian.
The new arrival was unable to influence the game before being replaced for Richarlison after an hour on the field, but it turns out he wasn't needed in the end.
Centre-back Van de Ven converted into an offensive threat to be feared.
Spurs' opening strike came from a well-worked corner routine, with Rodrigo Bentancur directing the ball toward the face of goal for an unmarked the defender to score.
And the captain - filling in as captain in Cristian Romero's unavailability - added his second in first-half stoppage time as he outjumped the England keeper to head home.
Van de Ven was equally outstanding in defensive duties, combining well with his defensive colleague as the pair handled superbly with the numerous deliveries thrown their way by the opposition.
Everton's recent stadium had been a happy hunting ground, but it appears the freshness may have worn off slightly and fans headed to the exits 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 striker situation will be.
Beto and Thierno Barry - the Everton's two established centre-forwards - had scored only one league goal between them before Sunday's outing.
And they were failed to add to that poor record on a damp and dreary night on Merseyside.
Beto's form was epitomized when he opted to bicycle-kick a clear chance from close range, despite having space and opportunity to reposition for a more conventional attempt that would have challenged Vicario.
Barry, too, failed to make an impact after coming on for the starter in the second half.
It was not for a lack of chance. Most of Everton's offensive moves came through Grealish and the forward, who relentlessly whipped dangerous crosses into the Spurs penalty area.
Everton: Moyes' men head to Sunderland in the top flight on early November at 20:00 GMT.
Tottenham: The London club travel to Newcastle's ground on late October for their fourth-round League Cup tie against the Magpies.
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Andrew Day
Andrew Day