Micky van de Ven became the first backline player to net two goals in a English top-flight match for Spurs since Jan Vertonghen in March 2013.
Key points
The Spurs defender heads home twice in opening period
The home side's Jake O'Brien has equalizer disallowed after VAR review
Pape Matar Sarr scores final goal in 89th minute
This loss is Everton's initial at Hill Dickinson Stadium
Micky van de Ven found the net on two occasions as Tottenham extended their unbeaten run on the road in the Premier League and delivered the Toffees their initial loss at their new home.
The Toffees were undefeated in their five previous fixtures at their recently opened stadium at the dockside location, having left their historic home behind after 133 years at the conclusion of last season.
But Tottenham's leader Van de Ven struck twice from set-pieces in the opening 45 minutes to place their impressive record at risk, and they were unable to turn it around before Pape Matar Sarr - assisted by former Toffees attacker the Brazilian - nodded in a third to seal the victory.
The Everton defender believed he had equalized the tally five minutes after Van de Ven's first goal, directing a bullet header past the Spurs keeper from an set-piece.
Beto, meanwhile, failed to convert a number of chances, while the Italian goalkeeper made a series of excellent stops to ensure Tottenham went five matches on the road unbeaten in the English top flight.
But the VAR intervened, instructing Craig Pawson to review the score on the pitchside monitor and the referee ruled that the winger and Iliman Ndiaye were in an illegal position and had interfered with the Tottenham goalkeeper.
This victory moves the manager's team up to third in the Premier League, five points off local foes and table-toppers the Gunners.
Everton stay 14th and are half a dozen points ahead of the relegation zone.
After a drab midweek performance in the Champions League against Monaco, where they mustered a only attempts on target, Frank opted for Randal Kolo Muani in attack instead of former Toffees player Richarlison.
The new arrival was failed to make much of an impact before being substituted for Richarlison after an hour on the pitch, but it emerged he wasn't needed in the end.
Centre-back Van de Ven transformed into an offensive threat to be feared.
Tottenham's opening strike came from a well-worked set-piece play, with Rodrigo Bentancur heading the ball toward the face of goal for an free the defender to score.
And the captain - filling in as captain in Cristian Romero's unavailability - notched his next goal in first-half stoppage time as he outjumped Jordan Pickford to nod in.
The Dutchman was equally impressive in defence, partnering effectively with Kevin Danso as the pair dealt excellently with the numerous deliveries thrown their way by Everton.
Everton's recent stadium had been a fortress, but it seems the freshness may have faded somewhat and fans flocked to the exits after the midfielder's third for Tottenham.
The end in Everton's unbeaten home run won't be a massive worry for David Moyes, but their attacking woes are.
Beto and the other striker - the Everton's two established centre-forwards - had scored only one league goal between them before the weekend's outing.
And they were failed to contribute to that poor record on a wet and gloomy evening on Merseyside.
The striker's form was epitomized when he opted to bicycle-kick a golden opportunity from close range, even with having room and time to reposition for a simpler shot that would have challenged the keeper.
The substitute, too, was unable to influence the game after coming on for the starter in the second half.
It was not for a shortage of chance. Most of the home side's offensive moves came through the winger and Ndiaye, who relentlessly whipped dangerous crosses into the Tottenham penalty area.
Everton: The team head to Sunderland in the top flight on early November at 20:00 GMT.
Spurs: Spurs go to St James' Park on Wednesday, 29 October for their fourth-round Carabao Cup tie against the Magpies.
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