Micky van de Ven became the initial backline player to net two goals in a Premier League match for Tottenham since the Belgian defender in spring 2013.
Match summary
The Spurs defender heads home two times in first half
The home side's Jake O'Brien has goal disallowed after VAR review
Pape Matar Sarr nets third in 89th minute
This loss is the Merseyside club's first at their new ground
The Tottenham defender found the net twice as Spurs prolonged their unbeaten run on the road in the top division and handed Everton their first defeat at Hill Dickinson Stadium.
Everton were undefeated in their last five matches at their recently opened stadium at the dockside location, having left Goodison Park behind after 133 years at the end of the previous campaign.
But Tottenham's leader the Dutchman struck on two occasions from corners in the opening 45 minutes to place their impressive record at risk, and they were unable to turn it around before the midfielder - assisted by ex-Everton forward Richarlison - headed home a third to put the game to bed.
The Everton defender thought he had equalized the tally a short while after Van de Ven's first goal, directing a powerful header past the Spurs keeper from an set-piece.
Beto, meanwhile, missed a number of chances, while Vicario made a succession of excellent saves to ensure the London club went five matches on the road unbeaten in the English top flight.
But the video assistant referee intervened, advising Craig Pawson to check the score on the screen and the official ruled that Jack Grealish and the attacker were in offside positions and had interfered with the Spurs keeper.
This victory elevates Thomas Frank's team into third place in the Premier League, five points off north London rivals and table-toppers Arsenal.
The Merseyside club stay in 14th position and sit six points ahead of the relegation zone.
Following a drab weekday performance in the European competition against Monaco, where they mustered a only two shots on target, Frank chose Randal Kolo Muani in attack instead of former Toffees player Richarlison.
The summer signing was failed to make much of an impact before being substituted for the substitute after 60 minutes on the field, but it emerged he proved unnecessary in the end.
Defender Van de Ven converted into an attacking force to be reckoned with.
Tottenham's first goal came from a well-worked set-piece play, with the midfielder heading the ball back across the six-yard box for an unmarked Van de Ven to head home.
And the captain - stepping up as skipper in Cristian Romero's unavailability - notched his second in first-half stoppage time as he rose above Jordan Pickford to head home.
Van de Ven was just as outstanding in defensive duties, combining effectively with his defensive colleague as the duo dealt excellently with the numerous deliveries thrown their way by Everton.
Everton's recent stadium had been a happy hunting ground, but it seems the novelty may have faded somewhat and fans flocked to the exits after Sarr's third for Spurs.
The break in Everton's unbeaten home run will not be a massive concern for David Moyes, but their striker situation are.
The forward and Thierno Barry - the Toffees' pair of recognised strikers - had netted only one league goal combined before the weekend's match.
And they were failed to add to that miserable record on a damp and dreary night on in Liverpool.
Beto's performance was epitomized when he opted to overhead kick a clear chance from near the goal, despite having room and opportunity to reposition for a more conventional shot that would have challenged Vicario.
Barry, as well, failed to influence the game after replacing Beto in the second half.
It was not for a lack of opportunity. Most of the home side's attacks came through Grealish and the forward, who tirelessly whipped dangerous balls into the Spurs box.
The Toffees: The team head to Sunderland in the Premier League on Monday, 3 November at 8 PM.
Tottenham: The London club go to St James' Park on Wednesday, 29 October for their next round League Cup tie against Newcastle.
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