The Dutch defender became the initial defender to net two goals in a English top-flight game for Spurs since Jan Vertonghen in March 2013.
Match summary
Micky Van de Ven nods home two times in opening period
Everton's defender has goal disallowed after VAR review
Senegalese midfielder scores final goal in 89th minute
The defeat is the Merseyside club's initial at Hill Dickinson Stadium
The Tottenham defender found the net on two occasions as Spurs extended their winning streak in away games in the top division and handed Everton their initial loss at Hill Dickinson Stadium.
Everton were undefeated in their five previous fixtures at their new ground at the dockside location, having left their historic home behind after more than a century at the end of last season.
But Tottenham's leader the Dutchman struck on two occasions from set-pieces in the opening 45 minutes to put their strong form at risk, and they were unable to turn it around before the midfielder - set up by former Toffees attacker Richarlison - nodded in a third to seal the victory.
Jake O'Brien believed he had equalized the tally a short while after the defender's opener, glancing a powerful header past the Spurs keeper from an Everton corner.
The Portuguese striker, in the meantime, failed to convert a handful of chances, while the Italian goalkeeper made a series of fine saves to ensure the London club went five matches on the road unbeaten in the English top flight.
But the VAR stepped in, advising the match official to check the score on the screen and the official decided that the winger and Iliman Ndiaye were in offside positions and had interfered with the Spurs keeper.
The win elevates the manager's team up to third in the division, five points off north London rivals and leaders the Gunners.
Everton stay 14th and sit six points clear of the drop zone.
After a uninspiring weekday performance in the Champions League against Monaco, where they managed a only attempts on goal, Frank chose the French striker up front instead of ex-Everton forward Richarlison.
The summer signing was unable to make much of an impact before being substituted for the substitute after 60 minutes on the pitch, but it emerged he proved unnecessary in the final outcome.
Centre-back the Dutchman transformed into an attacking force to be reckoned with.
Tottenham's first goal 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 the captain - stepping up as skipper in Cristian Romero's absence - notched his second in first-half stoppage time as he outjumped Jordan Pickford to nod in.
Van de Ven was equally impressive in defence, partnering effectively with Kevin Danso as the duo dealt excellently with the numerous deliveries thrown their way by Everton.
The club's recent stadium had been a fortress, but it appears the freshness may have worn off somewhat and supporters headed to the exits after the midfielder's third for Tottenham.
The end in Everton's winning streak at home will not be a massive concern for the manager, but their attacking woes are.
Beto and Thierno Barry - the Toffees' pair of established centre-forwards - had netted only one league goal between them before the weekend's outing.
And they were failed to contribute to that poor record on a damp and dreary night on Merseyside.
Beto's form was epitomized when he chose to bicycle-kick a golden opportunity from near the goal, despite having space and opportunity to readjust for a simpler shot that would have tested the keeper.
The substitute, as well, was unable to influence the game after replacing the starter in the second half.
It was not for a lack of chance. The majority of Everton's offensive moves came through Grealish and Ndiaye, who relentlessly whipped threatening balls into the Spurs penalty area.
The Toffees: The team head to the Stadium of Light in the Premier League on early November at 8 PM.
Spurs: The London club go to Newcastle's ground on Wednesday, 29 October for their next round Carabao Cup tie against the Magpies.
Riley Vance is a passionate esports journalist with over five years of experience covering major gaming events and interviewing top players worldwide.
Lynn Anderson
Lynn Anderson
Lynn Anderson
Lynn Anderson