Micky van de Ven emerged as the first defender to score two goals in a English top-flight game for Spurs since the Belgian defender in spring 2013.
At a glance
The Spurs defender nods into the net two times in first half
Everton's defender has equalizer ruled after video check
Senegalese midfielder scores final goal in 89th minute
This loss is Everton's first at Hill Dickinson Stadium
The Tottenham defender found the net twice as Spurs prolonged their unbeaten run on the road in the top division and handed the Toffees their initial loss at their new home.
The Toffees were undefeated in their five previous fixtures at their new ground at Bramley-Moore Dock, having left Goodison Park behind after more than a century at the end of the previous campaign.
But Spurs captain Van de Ven pounced twice from set-pieces in the first half to put their impressive record at jeopardy, and they were failed to turn it around before the midfielder - set up by ex-Everton forward the Brazilian - nodded in a third to seal the victory.
The Everton defender thought he had equalized the tally a short while after the defender's first goal, glancing a powerful header past Guglielmo Vicario from an set-piece.
Beto, meanwhile, missed a number of chances, while Vicario made a succession of excellent stops to guarantee Tottenham went five away games without defeat in the Premier League.
But the VAR intervened, advising Craig Pawson to review the goal on the screen and the official ruled that the winger and Iliman Ndiaye were in an illegal position and had interfered with the Tottenham goalkeeper.
This victory moves Thomas Frank's team into third place in the Premier League, five points off north London rivals and leaders the Gunners.
Everton stay 14th and are six points clear of the drop zone.
Following a uninspiring weekday display in the Champions League against Monaco, where they mustered a mere attempts on goal, Frank chose the French striker in attack instead of former Toffees player Richarlison.
The summer signing was failed to influence the game before being replaced for Richarlison after 60 minutes on the field, but it turns out he proved unnecessary in the end.
Centre-back Van de Ven transformed into an offensive threat to be feared.
Spurs' opening strike came from a cleverly executed set-piece play, with the midfielder directing the ball back across the face of goal for an unmarked the defender to score.
And the captain - filling in as captain in the Argentine's absence - added his next goal 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 his defensive colleague as the duo dealt superbly with the 34 crosses sent their way by Everton.
The club's recent stadium had been a happy hunting ground, but it seems the novelty may have worn off somewhat and fans headed to the exits after Sarr's final goal for Tottenham.
The end in Everton's unbeaten home run won't be a massive concern for the manager, but their attacking woes will be.
The forward and the other striker - the Everton's two established centre-forwards - had netted only one league goal combined before the weekend's match.
And they were failed to contribute to that poor record on a damp and dreary night on Merseyside.
Beto's form 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 simpler attempt that would have tested the keeper.
Barry, too, was unable to influence the game after replacing the starter in the second half.
It wasn't for a lack of chance. Most of Everton's offensive moves came through the winger and Ndiaye, who relentlessly whipped threatening balls into the Spurs box.
Everton: The team head to Sunderland in the top flight on early November at 8 PM.
Spurs: The London club go to Newcastle's ground on late October for their fourth-round League Cup match against the Magpies.
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