Micky van de Ven became the initial backline player to score two goals in a English top-flight game for Spurs since Jan Vertonghen in March 2013.
Match summary
The Spurs defender heads home twice in first half
Everton's defender has goal ruled after video check
Senegalese midfielder scores final goal in dying moments
This loss is Everton's initial at their new ground
Micky van de Ven scored twice as Tottenham extended their unbeaten run in away games in the Premier League and handed the Toffees their initial loss at their new home.
The Toffees were unbeaten in their last five matches at their new ground at the dockside location, having left their historic home behind after more than a century at the end of the previous campaign.
But Tottenham's leader Van de Ven pounced twice from corners in the first half to place their impressive record at risk, and they were failed to turn it around before Pape Matar Sarr - assisted by former Toffees attacker the Brazilian - headed home a third to seal the victory.
The Everton defender thought he had equalized the tally a short while after Van de Ven's first goal, glancing a bullet header past Guglielmo Vicario from an Everton corner.
The Portuguese striker, in the meantime, missed a handful of opportunities, while Vicario made a series of fine stops to ensure the London club went five matches on the road without defeat in the English top flight.
But the video assistant referee stepped in, instructing the match official to review the score on the pitchside monitor and the referee decided that the winger and Iliman Ndiaye were in offside positions and had impeded with the Spurs keeper.
This victory elevates Thomas Frank's side into third place in the Premier League, five points off north London rivals and table-toppers the Gunners.
The Merseyside club stay 14th and sit six points clear of the drop zone.
Following a uninspiring midweek performance in the Champions League against Monaco, where they managed a only two shots on goal, the manager opted for the French striker in attack instead of ex-Everton forward the Brazilian.
The new arrival was unable to make much of an impact before being substituted for Richarlison after 60 minutes on the field, but it turns out he wasn't needed in the end.
Defender Van de Ven transformed into an offensive threat to be reckoned with.
Tottenham's opening strike came from a cleverly executed corner routine, with Rodrigo Bentancur heading the ball back across the six-yard box for an unmarked the defender to score.
And the captain - filling in as skipper in Cristian Romero's absence - added his next goal in first-half stoppage time as he outjumped Jordan Pickford to head home.
The Dutchman was just as outstanding in defence, combining effectively with his defensive colleague as the pair handled superbly with the 34 crosses thrown their way by Everton.
Everton's new home had been a happy hunting ground, but it appears the freshness may have worn off slightly and supporters headed to the gates after the midfielder's third for Spurs.
The end in Everton's unbeaten home run will not be a massive concern for the manager, but their striker situation are.
The forward and the other striker - the Everton's pair of established centre-forwards - had netted just a single goal between them before the weekend's match.
And they were unable to add to that poor record on a damp and dreary evening on in Liverpool.
Beto's form was epitomized when he chose to overhead kick a clear chance from near the goal, even with having space and opportunity to readjust for a more conventional attempt that would have tested Vicario.
Barry, too, failed to influence the game after replacing Beto in the second half.
It was not for a lack of chance. The majority of the home side's attacks came through Grealish and the forward, who relentlessly whipped dangerous crosses into the Tottenham box.
Everton: The team travel to Sunderland in the top flight on early November at 20:00 GMT.
Spurs: The London club travel to St James' Park on late October for their fourth-round Carabao Cup match against Newcastle.