Micky van de Ven became the initial defender to score twice in a Premier League match for Tottenham since Jan Vertonghen in March 2013.
Key points
The Spurs defender heads home two times in first half
The home side's defender has goal ruled after video check
Pape Matar Sarr scores third in 89th minute
This loss is the Merseyside club's first at Hill Dickinson Stadium
The Tottenham defender scored on two occasions as Spurs extended their winning streak on the road in the top division and handed the Toffees their first defeat at Hill Dickinson Stadium.
Everton were undefeated in their five previous fixtures at their new ground at Bramley-Moore Dock, having departed their historic home behind after 133 years at the end of the previous campaign.
But Tottenham's leader Van de Ven pounced on two occasions from corners in the first half to place their strong form at jeopardy, and they were unable to mount a comeback before the midfielder - assisted by ex-Everton forward Richarlison - headed home a final goal to seal the victory.
The Everton defender believed he had equalized the tally a short while after Van de Ven's opener, glancing a powerful header past Guglielmo Vicario from an set-piece.
Beto, in the meantime, failed to convert a number of opportunities, while the Italian goalkeeper made a series of fine saves to ensure the London club went five matches on the road without defeat in the Premier League.
But the video assistant referee stepped in, advising Craig Pawson to review the goal on the pitchside monitor and the referee decided that the winger and Iliman Ndiaye were in an illegal position and had impeded with the Spurs keeper.
This victory moves Thomas Frank's team into third place in the Premier League, a handful of points off north London rivals and leaders Arsenal.
The Merseyside club remain 14th and sit half a dozen points ahead of the drop zone.
Following a uninspiring midweek display in the Champions League against the French side, where they mustered a only two shots on goal, the manager chose Randal Kolo Muani 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 an hour on the field, but it turns out he proved unnecessary in the final outcome.
Centre-back Van de Ven converted into an offensive threat to be feared.
Tottenham's first goal came from a cleverly executed set-piece play, with the midfielder heading the ball toward the six-yard box for an free the defender to score.
And the captain - stepping up as skipper in Cristian Romero's absence - added his next goal in first-half stoppage time as he outjumped Jordan Pickford to nod in.
The Dutchman was just as outstanding in defensive duties, partnering well with his defensive colleague as the pair handled superbly with the 34 crosses thrown their way by the opposition.
The club's new home had been a fortress, but it appears the freshness may have worn off somewhat and supporters flocked to the gates after Sarr's third for Tottenham.
The end in Everton's unbeaten home run will not be a major worry for the manager, but their striker situation are.
The forward and the other striker - the Everton's two recognised centre-forwards - had scored only one league goal between them before Sunday's match.
And they were unable to contribute to that miserable tally on a damp and dreary night on Merseyside.
The striker's form was summed up when he chose to overhead kick a golden opportunity from near the goal, even with having room and opportunity to reposition for a more conventional shot that would have tested the keeper.
Barry, as well, was unable to make an impact after replacing the starter in the second half.
It was not for a lack of opportunity. The majority of Everton's attacks came through Grealish and the forward, who relentlessly delivered dangerous balls into the Tottenham box.
Everton: The team head to the Stadium of Light in the Premier League on Monday, 3 November at 8 PM.
Spurs: The London club go to St James' Park on late October for their next round Carabao Cup tie against the Magpies.
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