Micky van de Ven emerged as the initial backline player to score two goals in a Premier League match for Tottenham since the Belgian defender in spring 2013.
Match summary
The Spurs defender nods home two times in opening period
The home side's Jake O'Brien has equalizer disallowed after video check
Senegalese midfielder scores final goal in dying moments
The defeat is Everton's initial at their new ground
Micky van de Ven scored on two occasions as Spurs extended their unbeaten run in away games in the top division and handed Everton their initial loss at their new home.
The Toffees were unbeaten in their last five matches at their new ground at the dockside location, having departed Goodison Park behind after 133 years at the conclusion of the previous campaign.
But Spurs captain Van de Ven struck twice from corners in the first half to place their impressive record at jeopardy, and they were unable to mount a comeback before Pape Matar Sarr - assisted by ex-Everton forward Richarlison - headed home a third to put the game to bed.
The Everton defender believed he had equalized the scores a short while after Van de Ven's first goal, glancing a bullet header past the Spurs keeper from an Everton corner.
Beto, in the meantime, missed a handful of opportunities, while the Italian goalkeeper made a series of excellent stops to ensure the London club went five matches on the road without defeat in the Premier League.
But the VAR intervened, advising the match official to review the goal on the screen and the official ruled that Jack Grealish and the attacker were in an illegal position and had impeded with the Spurs keeper.
The win moves Thomas Frank's side up to third in the division, five points off local foes and table-toppers the Gunners.
Everton remain 14th and are half a dozen points ahead of the relegation zone.
After a drab weekday performance in the European competition against the French side, where they managed a only two shots on goal, the manager chose Randal Kolo Muani in attack instead of former Toffees player the Brazilian.
The summer signing was failed to influence the game before being replaced for the substitute after an hour on the pitch, but it turns out he wasn't needed in the end.
Centre-back Van de Ven transformed into an attacking force to be feared.
Tottenham's first goal came from a well-worked set-piece play, with the midfielder heading the ball toward the face of goal for an free Van de Ven to head home.
And Ven de Ven - stepping up as captain in the Argentine'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 defence, partnering effectively with his defensive colleague as the pair dealt excellently with the 34 crosses thrown their way by the opposition.
The club's new home had been a fortress, but it seems the freshness may have worn off slightly and fans flocked to the exits after Sarr's final goal for Spurs.
The break in the team's unbeaten home run won't be a major worry for the manager, but their attacking woes are.
Beto and Thierno Barry - the Everton's pair of recognised centre-forwards - had scored just a single goal between them before Sunday's outing.
And they were unable to add to that poor tally on a wet and gloomy evening on in Liverpool.
Beto's performance was epitomized when he opted to bicycle-kick a clear chance from near the goal, even with having space and opportunity to reposition for a simpler shot that would have tested Vicario.
The substitute, too, was unable to make an impact after replacing the starter in the 65th minute.
It was not for a lack of opportunity. Most of the home side's attacks came through the winger and the forward, who tirelessly whipped dangerous crosses into the Spurs box.
Everton: Moyes' men head to Sunderland in the Premier League on Monday, 3 November at 20:00 GMT.
Spurs: Spurs travel to St James' Park on Wednesday, 29 October for their next round Carabao Cup match against Newcastle.
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