Micky van de Ven became the initial defender to net twice in a Premier League match for Spurs since the Belgian defender in March 2013.
At a glance
The Spurs defender heads home two times in opening period
Everton's Jake O'Brien has equalizer ruled after video check
Senegalese midfielder scores third in 89th minute
This loss is Everton's initial at Hill Dickinson Stadium
The Tottenham defender scored on two occasions as Tottenham extended their winning streak in away games in the Premier League and handed Everton their initial loss at their new home.
The Toffees were undefeated in their last five matches at their recently opened stadium at Bramley-Moore Dock, having departed their historic home behind after 133 years at the conclusion of the previous campaign.
But Spurs captain Van de Ven pounced on two occasions from set-pieces in the opening 45 minutes to put their strong form at risk, and they were failed to mount a comeback before Pape Matar Sarr - assisted by ex-Everton forward the Brazilian - nodded in a third to seal the victory.
The Everton defender believed he had levelled the scores a short while after Van de Ven's first goal, glancing a powerful header past Guglielmo Vicario from an set-piece.
The Portuguese striker, meanwhile, missed a handful of opportunities, while Vicario made a succession of excellent saves to guarantee Tottenham went five away games unbeaten in the English top flight.
But the video assistant referee stepped in, instructing the match official to check the score on the pitchside monitor and the referee ruled that the winger and the attacker were in offside positions and had interfered with the Spurs keeper.
The win elevates Thomas Frank's side up to third in the division, a handful of points off north London rivals and leaders the Gunners.
Everton stay in 14th position and sit half a dozen points ahead of the drop zone.
After a drab weekday display in the Champions League against Monaco, where they mustered a mere two shots on goal, the manager chose the French striker in attack instead of former Toffees player Richarlison.
The new arrival was unable to influence the game before being substituted for the substitute after 60 minutes on the field, but it emerged he proved unnecessary in the end.
Centre-back Van de Ven converted into an offensive threat to be feared.
Tottenham's opening strike came from a well-worked set-piece play, with Rodrigo Bentancur directing the ball toward the six-yard box for an unmarked Van de Ven to score.
And the captain - stepping up as skipper in Cristian Romero's unavailability - added his next goal in added time as he rose above Jordan Pickford to head home.
The Dutchman was equally outstanding in defensive duties, partnering well with Kevin Danso as the duo dealt excellently with the numerous deliveries thrown their way by the opposition.
The club's new home had been a happy hunting ground, but it appears the freshness may have worn off somewhat and supporters headed to the exits after Sarr's final goal for Spurs.
The end in the team's unbeaten home run will not be a massive worry for David Moyes, but their striker situation are.
The forward and Thierno Barry - the Everton's two recognised strikers - had netted only one league goal between them before the weekend's match.
And they were failed to add to that miserable record on a wet and gloomy evening on Merseyside.
Beto's form was summed up when he opted to overhead kick a clear chance from close range, even with having room and time to reposition for a simpler shot that would have challenged Vicario.
Barry, as well, failed to influence the game after coming on for the starter in the second half.
It wasn't for a lack of opportunity. The majority of Everton's attacks came through Grealish and the forward, who relentlessly delivered dangerous balls into the Tottenham box.
The Toffees: Moyes' men head to Sunderland in the Premier League on early November at 20:00 GMT.
Spurs: Spurs go to Newcastle's ground on late October for their next round League Cup match against Newcastle.
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