The Dutch defender emerged as the first backline player to net twice in a Premier League game for Tottenham since the Belgian defender in March 2013.
Key points
The Spurs defender heads home twice in opening period
Everton's defender has goal ruled after video check
Senegalese midfielder scores final goal in 89th minute
The defeat is Everton's first at their new ground
Micky van de Ven scored on two occasions as Tottenham prolonged their winning streak in away games in the top division and delivered the Toffees their initial loss at Hill Dickinson Stadium.
Everton were undefeated in their five previous fixtures at their recently opened stadium at Bramley-Moore Dock, having left Goodison Park behind after more than a century at the end of last season.
But Spurs captain the Dutchman pounced on two occasions from set-pieces in the first half 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 put the game to bed.
The Everton defender thought he had levelled the scores five minutes after the defender's first goal, directing a bullet header past Guglielmo Vicario from an set-piece.
The Portuguese striker, meanwhile, missed a number of opportunities, while Vicario made a succession of fine saves to ensure 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 decided that Jack Grealish and the attacker were in an illegal position and had interfered with the Tottenham goalkeeper.
The win moves the manager's team into third place in the Premier League, a handful of points off local foes and table-toppers Arsenal.
Everton stay in 14th position and sit half a dozen points clear of the relegation zone.
Following a uninspiring midweek performance in the Champions League against the French side, where they mustered a only attempts on goal, Frank chose the French striker in attack instead of ex-Everton forward the Brazilian.
The new arrival was unable to make much of an impact before being replaced for Richarlison after an hour on the pitch, but it emerged he wasn't needed in the final outcome.
Centre-back Van de Ven transformed into an attacking force to be reckoned with.
Tottenham's first goal came from a cleverly executed corner routine, with Rodrigo Bentancur directing the ball toward the six-yard box for an free Van de Ven to score.
And the captain - stepping up as captain in Cristian Romero's absence - 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, partnering well with Kevin Danso as the duo handled excellently with the numerous deliveries sent their way by Everton.
Everton's recent stadium had been a happy hunting ground, but it appears the novelty may have faded somewhat and fans headed to the gates after the midfielder's final goal for Tottenham.
The end in the team's winning streak at home will not be a major worry for the manager, but their striker situation are.
The forward and the other striker - the Everton's pair of established centre-forwards - had netted just one league goal combined before the weekend's outing.
And they were unable to contribute to that poor tally on a damp and dreary evening on in Liverpool.
Beto's performance was epitomized when he chose to bicycle-kick a golden opportunity from close range, even with having space and time to reposition for a simpler attempt that would have challenged Vicario.
The substitute, too, failed to make an impact after replacing the starter in the second half.
It was not for a lack of opportunity. Most of Everton's offensive moves came through Grealish and Ndiaye, who relentlessly delivered threatening balls into the Tottenham penalty area.
Everton: Moyes' men travel to Sunderland in the top flight on early November at 8 PM.
Tottenham: Spurs travel to Newcastle's ground on Wednesday, 29 October for their next round Carabao Cup tie against Newcastle.