The Dutch defender emerged as the first defender to score twice in a English top-flight game for Tottenham since Jan Vertonghen in March 2013.
At a glance
The Spurs defender heads into the net two times in opening period
Everton's defender has equalizer ruled after video check
Pape Matar Sarr scores third in 89th minute
This loss is the Merseyside club's initial at Hill Dickinson Stadium
Micky van de Ven scored twice as Spurs extended their unbeaten run in away games in the Premier League and handed Everton their first defeat at their new home.
The Toffees were unbeaten in their five previous fixtures at their recently opened stadium at the dockside location, having departed Goodison Park behind after 133 years at the conclusion of last season.
But Tottenham's leader the Dutchman pounced twice from set-pieces in the first half to put their strong form at risk, and they were failed to turn it around before the midfielder - set up by ex-Everton forward Richarlison - headed home a third to put the game to bed.
The Everton defender believed he had equalized the tally five minutes after Van de Ven's opener, directing a bullet header past Guglielmo Vicario from an Everton corner.
Beto, in the meantime, missed a number of opportunities, while the Italian goalkeeper made a succession of fine saves to guarantee the London club went five matches on the road unbeaten in the English top flight.
But the video assistant referee stepped in, advising the match official to check the score on the screen and the referee ruled that Jack Grealish and the attacker were in offside positions and had interfered with the Tottenham goalkeeper.
The win moves Thomas Frank's team into third place in the division, five points off north London rivals and leaders Arsenal.
Everton remain 14th and sit half a dozen points clear of the relegation zone.
After a uninspiring midweek display in the European competition against Monaco, where they mustered a mere two shots on goal, Frank chose Randal Kolo Muani in attack instead of ex-Everton forward the Brazilian.
The summer signing was unable to make much of an impact before being substituted for the substitute after 60 minutes on the field, but it emerged he wasn't needed in the final outcome.
Centre-back the Dutchman converted into an offensive threat to be feared.
Spurs' first goal came from a well-worked set-piece play, with the midfielder heading the ball back across the six-yard box for an free Van de Ven to head home.
And the captain - filling in as captain in Cristian Romero's absence - notched his second in first-half stoppage time as he rose above Jordan Pickford to head home.
Van de Ven was just as impressive in defence, combining effectively with his defensive colleague as the pair handled excellently with the 34 crosses sent their way by Everton.
Everton's new home had been a fortress, but it seems the freshness may have faded somewhat and supporters headed to the gates after the midfielder's final goal for Spurs.
The break in Everton's unbeaten home run will not be a massive worry for David Moyes, but their striker situation are.
The forward and the other striker - the Toffees' two established centre-forwards - had scored only one league goal combined before the weekend's outing.
And they were failed to contribute to that poor tally on a wet and gloomy evening on Merseyside.
The striker's form was epitomized when he opted to overhead kick a golden opportunity from near the goal, despite having space and opportunity to reposition for a more conventional attempt that would have challenged the keeper.
Barry, too, failed to make an impact after replacing Beto in the 65th minute.
It wasn't for a lack of chance. The majority of Everton's offensive moves came through Grealish and the forward, who relentlessly delivered threatening crosses into the Tottenham penalty area.
The Toffees: Moyes' men travel to Sunderland in the Premier League on Monday, 3 November at 8 PM.
Spurs: The London club travel to St James' Park on late October for their next round Carabao Cup tie against Newcastle.
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