Micky van de Ven became the first backline player to net two goals in a English top-flight match for Spurs since the Belgian defender in March 2013.
Key points
The Spurs defender heads into the net two times in opening period
Everton's Jake O'Brien has goal ruled after video check
Senegalese midfielder nets third in 89th minute
The defeat is Everton's initial at their new ground
The Tottenham defender scored on two occasions as Tottenham extended their winning streak on the road in the Premier League and delivered Everton their first defeat at Hill Dickinson Stadium.
The Toffees were undefeated in their five previous fixtures at their recently opened stadium at the dockside location, having departed Goodison Park behind after more than a century at the end of last season.
But Spurs captain Van de Ven pounced twice from set-pieces 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 the Brazilian - headed home a third to seal the victory.
Jake O'Brien thought he had levelled the scores five minutes after Van de Ven's opener, directing a powerful header past the Spurs keeper from an set-piece.
Beto, meanwhile, failed to convert a number of opportunities, while the Italian goalkeeper made a succession of excellent stops to guarantee the London club went five away games without defeat in the Premier League.
But the VAR intervened, advising the match official to review the goal on the screen and the official decided that the winger and the attacker were in offside positions and had interfered with the Spurs keeper.
The win moves the manager's side up to third in the Premier League, a handful of points off north London rivals and table-toppers the Gunners.
The Merseyside club remain in 14th position and sit six points ahead of the relegation zone.
After a drab midweek performance in the Champions League against the French side, where they managed a only two shots on goal, Frank opted for Randal Kolo Muani in attack instead of ex-Everton forward the Brazilian.
The summer signing was unable to make much of an impact before being replaced for the substitute after an hour on the field, but it emerged he wasn't needed in the end.
Defender the Dutchman converted into an offensive threat to be feared.
Spurs' first goal came from a cleverly executed set-piece play, with the midfielder heading the ball back across the face of goal for an free Van de Ven to score.
And the captain - stepping up as skipper in Cristian Romero's absence - notched his second in first-half stoppage time as he outjumped Jordan Pickford to nod in.
Van de Ven was just as outstanding in defence, partnering effectively with Kevin Danso as the duo dealt superbly with the numerous deliveries sent their way by Everton.
The club's new home had been a fortress, but it seems the freshness may have faded somewhat and supporters flocked to the gates after Sarr's final goal for Tottenham.
The end in Everton's winning streak at home won't be a massive concern for David Moyes, but their striker situation are.
The forward and the other striker - the Everton's two recognised centre-forwards - had scored only one league goal between them before the weekend's match.
And they were unable to contribute to that poor record on a wet and gloomy night on Merseyside.
Beto's form was summed up when he opted to bicycle-kick a golden opportunity from close range, despite having room and time to reposition for a more conventional shot that would have tested Vicario.
Barry, too, 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 the winger and Ndiaye, who relentlessly delivered threatening crosses into the Spurs box.
The Toffees: The team head to the Stadium of Light in the top flight on early November at 8 PM.
Tottenham: The London club travel to Newcastle's ground on Wednesday, 29 October for their fourth-round Carabao Cup match against the Magpies.
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