Micky van de Ven became the first backline player to score twice in a Premier League match for Tottenham since Jan Vertonghen in spring 2013.
Key points
The Spurs defender nods home two times in first half
The home side's defender has goal disallowed after video check
Senegalese midfielder scores final goal in dying moments
The defeat is the Merseyside club's first at Hill Dickinson Stadium
Micky van de Ven scored twice as Tottenham prolonged their winning streak in away games in the top division and delivered Everton their initial loss at their new home.
The Toffees were undefeated in their last five matches at their new ground at Bramley-Moore Dock, having left their historic home behind after 133 years at the conclusion of last season.
But Tottenham's leader the Dutchman struck twice from corners in the first half to place their impressive record at risk, and they were unable to mount a comeback before Pape Matar Sarr - set up by former Toffees attacker Richarlison - headed home a third to put the game to bed.
Jake O'Brien thought he had levelled the tally five minutes after Van de Ven's first goal, glancing a powerful header past the Spurs keeper from an Everton corner.
Beto, meanwhile, missed a number of opportunities, while the Italian goalkeeper made a succession of excellent saves to guarantee the London club went five away games without defeat in the English top flight.
But the video assistant referee intervened, advising the match official to check the score on the pitchside monitor and the official ruled that the winger and Iliman Ndiaye were in offside positions and had interfered with the Tottenham goalkeeper.
The win moves the manager's side into third place in the Premier League, five points off local foes and table-toppers Arsenal.
Everton remain in 14th position and sit half a dozen points clear of the relegation zone.
After a drab weekday performance in the European competition against Monaco, where they managed a mere attempts on goal, Frank chose the French striker up front instead of former Toffees player Richarlison.
The summer signing was unable to influence the game before being substituted for Richarlison after 60 minutes on the pitch, but it turns out he wasn't needed in the final outcome.
Defender Van de Ven converted into an offensive threat to be reckoned with.
Tottenham's opening strike came from a well-worked corner routine, with Rodrigo Bentancur directing the ball back across the face of goal for an unmarked Van de Ven to score.
And Ven de Ven - filling in as skipper in Cristian Romero's absence - notched his next goal in first-half stoppage time as he outjumped Jordan Pickford to nod in.
Van de Ven was equally impressive in defence, partnering effectively with Kevin Danso as the pair handled superbly with the 34 crosses thrown their way by Everton.
Everton's recent stadium had been a fortress, but it seems the novelty may have faded slightly and fans headed to the exits after the midfielder's final goal for Spurs.
The break in the team's unbeaten home run won't be a major concern for the manager, but their striker situation are.
The forward and Thierno Barry - the Toffees' pair of established centre-forwards - had netted just one league goal combined before the weekend's match.
And they were failed to add to that poor record on a wet and gloomy night on in Liverpool.
Beto's performance was epitomized when he chose to overhead kick a golden opportunity from near the goal, even with having space and opportunity to readjust for a more conventional attempt that would have challenged the keeper.
The substitute, as well, was unable 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 tirelessly whipped threatening crosses into the Spurs penalty area.
The Toffees: The team travel to Sunderland in the Premier League on Monday, 3 November at 20:00 GMT.
Tottenham: Spurs go to St James' Park on late October for their next round League Cup match against Newcastle.
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Dr. Matthew Thomas Jr.
Dr. Matthew Thomas Jr.
Dr. Matthew Thomas Jr.