The Dutch defender emerged as the first backline player to net twice in a English top-flight game for Tottenham since the Belgian defender in March 2013.
At a glance
The Spurs defender nods into the net twice in opening period
The home side's defender has equalizer disallowed after video check
Senegalese midfielder scores final goal in 89th minute
The defeat is the Merseyside club's first at Hill Dickinson Stadium
Micky van de Ven found the net on two occasions as Spurs prolonged their winning streak on the road in the top division and delivered Everton their initial loss at their new home.
The Toffees were undefeated in their five previous fixtures at their new ground at Bramley-Moore Dock, having left Goodison Park behind after 133 years at the end of last season.
But Spurs captain Van de Ven struck on two occasions from set-pieces in the opening 45 minutes to place their impressive record at jeopardy, and they were failed to mount a comeback before Pape Matar Sarr - set up by former Toffees attacker Richarlison - headed home a third to seal the victory.
Jake O'Brien thought he had equalized the tally a short while after the defender's opener, directing a bullet header past Guglielmo Vicario from an set-piece.
The Portuguese striker, meanwhile, failed to convert a number of opportunities, while the Italian goalkeeper made a succession of fine stops to ensure the London club went five away games unbeaten in the Premier League.
But the VAR stepped in, advising Craig Pawson to check the score on the screen and the referee decided that the winger and the attacker were in offside positions and had interfered with the Spurs keeper.
The win moves the manager's side up to third in the division, a handful of points off local foes and leaders the Gunners.
The Merseyside club remain in 14th position and are six points clear of the relegation zone.
Following a drab weekday display in the European competition against the French side, where they mustered a mere two shots on target, Frank opted for Randal Kolo Muani in attack instead of ex-Everton forward Richarlison.
The summer signing was unable to influence the game before being replaced for the substitute after 60 minutes on the pitch, but it emerged he proved unnecessary in the end.
Defender Van de Ven transformed into an offensive threat to be feared.
Spurs' opening strike came from a well-worked corner routine, with Rodrigo Bentancur directing the ball back across the six-yard box for an free the defender to head home.
And the captain - filling in as skipper in Cristian Romero's unavailability - added his next goal in first-half stoppage time as he outjumped Jordan Pickford to head home.
The Dutchman was just as impressive in defence, combining well with his defensive colleague as the pair handled excellently with the 34 crosses thrown their way by the opposition.
Everton's recent stadium had been a happy hunting ground, but it seems the novelty may have worn off slightly and supporters headed to the gates after Sarr's final goal for Spurs.
The break in Everton's winning streak at home won't be a major worry for David Moyes, but their striker situation are.
The forward and the other striker - the Everton's pair of recognised centre-forwards - had scored just one league goal combined before Sunday's match.
And they were unable to add to that poor tally on a wet and gloomy evening on in Liverpool.
The striker's form was summed up when he chose to bicycle-kick a golden opportunity from close range, even with having room and time to reposition for a simpler attempt that would have challenged the keeper.
The substitute, too, failed to make an impact after coming on for Beto in the 65th minute.
It wasn't for a shortage of chance. The majority of Everton's attacks came through Grealish and Ndiaye, who tirelessly whipped threatening balls into the Spurs penalty area.
The Toffees: The team travel to the Stadium of Light in the Premier League on early November at 8 PM.
Tottenham: Spurs go to St James' Park on Wednesday, 29 October for their fourth-round League Cup match against Newcastle.
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