The Dutch defender became the first backline player to net twice in a English top-flight match for Spurs since the Belgian defender in spring 2013.
Key points
The Spurs defender heads into the net two times in first half
Everton's Jake O'Brien has goal ruled after video check
Senegalese midfielder scores third in dying moments
The defeat is Everton's initial at Hill Dickinson Stadium
Micky van de Ven found the net twice as Tottenham extended their winning streak on the road in the Premier League and delivered Everton their initial loss at Hill Dickinson Stadium.
The Toffees were unbeaten in their last five matches at their recently opened stadium at the dockside location, having departed their historic home behind after 133 years at the conclusion of the previous campaign.
But Tottenham's leader the Dutchman struck on two occasions from set-pieces in the opening 45 minutes to place their impressive record at risk, and they were unable to mount a comeback before Pape Matar Sarr - set up by former Toffees attacker the Brazilian - headed home a final goal to seal the victory.
Jake O'Brien believed he had equalized the tally a short while after Van de Ven's opener, glancing a bullet header past the Spurs keeper from an Everton corner.
Beto, meanwhile, failed to convert a number of opportunities, while Vicario made a series of excellent stops to guarantee Tottenham went five away games without defeat in the English top flight.
But the video assistant referee intervened, advising the match official to check the goal on the screen and the referee decided that Jack Grealish and Iliman Ndiaye were in an illegal position and had interfered with the Tottenham goalkeeper.
The win elevates the manager's team into third place in the division, five points off north London rivals and leaders the Gunners.
The Merseyside club stay 14th and are six points ahead of the drop zone.
After a uninspiring weekday performance in the European competition against the French side, where they mustered a mere two shots on target, the manager opted for the French striker in attack instead of ex-Everton forward Richarlison.
The new arrival was failed to influence the game before being replaced for Richarlison after an hour on the field, but it emerged he wasn't needed in the end.
Defender Van de Ven transformed into an attacking force to be reckoned with.
Tottenham's opening strike came from a well-worked corner routine, with Rodrigo Bentancur heading the ball toward the six-yard box for an unmarked Van de Ven to head home.
And Ven de Ven - filling in as skipper in Cristian Romero's absence - added his second in first-half stoppage time as he rose above the England keeper to nod in.
Van de Ven was equally outstanding in defensive duties, partnering effectively with his defensive colleague as the pair dealt superbly with the numerous deliveries sent their way by Everton.
The club's recent stadium had been a happy hunting ground, but it appears the novelty may have worn off somewhat and fans flocked to the exits after the midfielder's final goal for Tottenham.
The break in the team's unbeaten home run won't be a major worry for the manager, but their striker situation will be.
Beto and the other striker - the Everton's two established centre-forwards - had scored just one league goal combined before Sunday's outing.
And they were unable to add to that poor record on a wet and gloomy night on in Liverpool.
The striker's form was summed up when he chose to overhead kick a clear chance from close range, even with having room and time to reposition for a simpler attempt that would have challenged the keeper.
The substitute, too, was unable to make an impact after coming on for Beto in the second half.
It was not for a lack of opportunity. The majority of the home side's attacks came through the winger and the forward, who tirelessly whipped dangerous balls into the Spurs box.
Everton: The team travel to Sunderland in the top flight on early November at 20:00 GMT.
Tottenham: The London club go to Newcastle's ground on Wednesday, 29 October for their fourth-round Carabao Cup match against Newcastle.
Elena Vance is a logistics consultant with over a decade of experience in transport management and supply chain optimization.