Micky van de Ven became the initial defender to net twice in a English top-flight match for Spurs since the Belgian defender in March 2013.
Match summary
The Spurs defender heads home two times in first half
The home side's Jake O'Brien has equalizer ruled after VAR review
Pape Matar Sarr scores third in 89th minute
The defeat is the Merseyside club's first at their new ground
The Tottenham defender found the net twice as Spurs extended their unbeaten run in away games in the top division and handed Everton their first defeat at Hill Dickinson Stadium.
The Toffees were unbeaten in their five previous fixtures at their recently opened stadium at Bramley-Moore Dock, having departed Goodison Park behind after 133 years at the conclusion of the previous campaign.
But Tottenham's leader Van de Ven pounced on two occasions from set-pieces in the first half to place their impressive record at risk, and they were failed to turn it around before Pape Matar Sarr - set up by former Toffees attacker the Brazilian - nodded in a final goal to put the game to bed.
The Everton defender thought he had equalized the tally five minutes after Van de Ven's opener, glancing a bullet header past the Spurs keeper from an Everton corner.
Beto, meanwhile, failed to convert a handful of chances, while the Italian goalkeeper made a succession of excellent stops to guarantee Tottenham went five matches on the road without defeat in the English top flight.
But the VAR intervened, instructing the match official to check the score on the pitchside monitor and the referee decided that Jack Grealish and Iliman Ndiaye were in an illegal position and had interfered with the Spurs keeper.
The win moves the manager's team up to third in the Premier League, five points off north London rivals and table-toppers Arsenal.
Everton stay 14th and sit six points clear of the relegation zone.
Following a drab weekday display in the European competition against the French side, where they mustered a mere attempts on goal, the manager opted for Randal Kolo Muani in attack instead of ex-Everton forward the Brazilian.
The new arrival was failed to make much of an impact before being substituted for the substitute after an hour on the pitch, but it emerged he wasn't needed in the final outcome.
Defender the Dutchman converted into an offensive threat to be feared.
Spurs' opening strike came from a cleverly executed set-piece play, with Rodrigo Bentancur directing the ball back across the face of goal for an unmarked Van de Ven to score.
And Ven de Ven - stepping up as captain in Cristian Romero's absence - notched his second in added time as he outjumped the England keeper to nod in.
The Dutchman was equally outstanding in defence, partnering effectively with his defensive colleague as the duo handled excellently with the numerous deliveries sent their way by the opposition.
The club's recent stadium had been a happy hunting ground, but it appears the novelty may have worn off somewhat and fans headed to the gates after the midfielder's third for Spurs.
The break in the team's unbeaten home run will not be a massive concern for David Moyes, but their striker situation will be.
The forward and Thierno Barry - the Toffees' two recognised strikers - had netted only a single goal combined before the weekend's outing.
And they were unable to contribute to that poor record on a damp and dreary evening 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 reposition for a simpler shot that would have challenged Vicario.
The substitute, as well, failed to make an impact after coming on for the starter in the 65th minute.
It wasn't for a lack of chance. The majority of the home side's offensive moves came through Grealish and Ndiaye, who tirelessly delivered threatening crosses into the Tottenham box.
The Toffees: Moyes' men head to Sunderland in the Premier League on Monday, 3 November at 8 PM.
Tottenham: The London club travel to St James' Park on Wednesday, 29 October for their next round Carabao Cup match against Newcastle.
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