The Dutch defender emerged as the initial backline player to score two goals in a Premier League game for Spurs since the Belgian defender in March 2013.
Key points
The Spurs defender heads into the net twice in first half
Everton's defender has equalizer ruled after video check
Senegalese midfielder scores third in dying moments
The defeat is the Merseyside club's initial at Hill Dickinson Stadium
Micky van de Ven scored on two occasions as Tottenham prolonged their unbeaten run on the road in the top division and delivered the Toffees their first defeat at Hill Dickinson Stadium.
The Toffees were unbeaten in their five previous fixtures at their new ground at Bramley-Moore Dock, having departed Goodison Park behind after 133 years at the conclusion of the previous campaign.
But Spurs captain the Dutchman pounced on two occasions from set-pieces in the opening 45 minutes to put their impressive record at jeopardy, and they were failed to mount a comeback before Pape Matar Sarr - set up by ex-Everton forward the Brazilian - nodded in a final goal to seal the victory.
Jake O'Brien thought he had levelled the scores a short while after Van de Ven's opener, glancing a powerful header past the Spurs keeper from an set-piece.
Beto, meanwhile, missed a number of opportunities, while the Italian goalkeeper made a succession of excellent stops to ensure the London club went five away games unbeaten in the Premier League.
But the video assistant referee stepped in, advising Craig Pawson to check the score on the pitchside monitor and the referee decided that Jack Grealish and the attacker were in an illegal position and had impeded with the Spurs keeper.
The win moves Thomas Frank's team into third place in the Premier League, a handful of points off north London rivals and leaders Arsenal.
Everton remain in 14th position and are half a dozen points clear of the drop zone.
Following a uninspiring weekday performance in the European competition against the French side, where they mustered a only two shots on goal, Frank chose the French striker in attack instead of former Toffees player Richarlison.
The new arrival was failed to make much of an impact before being substituted for Richarlison after an hour on the field, but it emerged he wasn't needed in the final outcome.
Centre-back the Dutchman transformed into an attacking force to be feared.
Tottenham's 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 score.
And the captain - filling in as skipper in the Argentine's unavailability - notched his second in added time as he outjumped the England keeper to head home.
The Dutchman was just as impressive in defensive duties, combining effectively with his defensive colleague as the pair handled excellently with the 34 crosses sent their way by the opposition.
The club's recent stadium had been a happy hunting ground, but it seems the freshness may have worn off somewhat and fans headed to the exits after the midfielder's third for Spurs.
The end in the team's winning streak at home will not be a major concern for the manager, but their attacking woes are.
The forward and Thierno Barry - the Toffees' pair of established centre-forwards - had netted only one league goal combined before the weekend's outing.
And they were unable to add to that miserable record on a wet and gloomy night on Merseyside.
The striker's form was summed up when he opted to overhead kick a clear chance from near the goal, even with having space and opportunity to readjust for a simpler shot that would have challenged the keeper.
Barry, too, failed to make an impact after coming on for Beto in the second half.
It wasn't for a lack of chance. Most of the home side's offensive moves came through the winger and Ndiaye, who tirelessly delivered dangerous crosses into the Spurs penalty area.
The Toffees: Moyes' men head to Sunderland in the Premier League on Monday, 3 November at 20:00 GMT.
Spurs: The London club travel to Newcastle's ground on Wednesday, 29 October for their next round Carabao Cup tie against Newcastle.
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