The Dutch defender emerged as the first defender to net two goals in a Premier League match for Tottenham since the Belgian defender in March 2013.
Key points
Micky Van de Ven heads home two times in opening period
The home side's Jake O'Brien has equalizer disallowed after video check
Pape Matar Sarr nets final goal in dying moments
This loss is Everton's initial at their new ground
The Tottenham defender scored on two occasions as Tottenham extended their winning streak on the road in the Premier League and delivered the Toffees their initial loss at Hill Dickinson Stadium.
The Toffees were unbeaten in their five previous fixtures at their recently opened stadium at the dockside location, having left Goodison Park behind after more than a century at the end of last season.
But Spurs captain the Dutchman pounced twice from corners in the opening 45 minutes to place their impressive record at jeopardy, and they were failed to mount a comeback before the midfielder - assisted by former Toffees attacker Richarlison - nodded in a final goal to put the game to bed.
Jake O'Brien believed he had equalized the tally a short while after the defender's opener, glancing a bullet header past the Spurs keeper from an set-piece.
The Portuguese striker, meanwhile, failed to convert a handful of opportunities, while the Italian goalkeeper made a series of excellent saves to ensure Tottenham went five away games without defeat in the English top flight.
But the video assistant referee stepped in, instructing Craig Pawson to review the score on the screen and the official decided that Jack Grealish and the attacker were in offside positions and had impeded with the Tottenham goalkeeper.
This victory elevates the manager's side into third place in the division, a handful of points off north London rivals and leaders Arsenal.
The Merseyside club stay in 14th position and are six points clear of the drop zone.
After a uninspiring midweek performance in the European competition against Monaco, where they mustered a only two shots on goal, the manager opted for the French striker up front instead of former Toffees player the Brazilian.
The new arrival was unable to influence the game before being replaced for the substitute after 60 minutes on the field, but it emerged he wasn't needed in the end.
Defender the Dutchman converted into an attacking force to be reckoned with.
Spurs' opening strike came from a cleverly executed set-piece play, with Rodrigo Bentancur heading the ball toward the face of goal for an unmarked Van de Ven to head home.
And Ven de Ven - stepping up as skipper in Cristian Romero's unavailability - added his second in added time as he rose above the England keeper to head home.
The Dutchman was equally impressive in defensive duties, partnering effectively with Kevin Danso as the pair handled superbly with the 34 crosses sent their way by the opposition.
Everton's recent stadium had been a happy hunting ground, but it appears the freshness may have faded slightly and supporters headed to the exits after Sarr's third for Tottenham.
The end in Everton's unbeaten home run will not be a major worry for the manager, but their attacking woes will be.
Beto and the other striker - the Everton's pair of established strikers - had scored only one league goal combined before Sunday's match.
And they were unable to contribute to that miserable tally on a wet and gloomy night on in Liverpool.
Beto's form was summed up when he chose to overhead kick a clear chance from near the goal, despite having space and time to reposition for a simpler attempt that would have tested the keeper.
The substitute, too, failed to influence the game after replacing Beto in the 65th minute.
It was not for a shortage of chance. Most of Everton's offensive moves came through the winger and the forward, who tirelessly whipped dangerous balls into the Tottenham penalty area.
Everton: Moyes' men head to the Stadium of Light in the Premier League on early November at 20:00 GMT.
Tottenham: The London club travel to Newcastle's ground on late October for their next round League Cup match against the Magpies.