Micky van de Ven became the initial defender to net two goals in a Premier League match for Spurs since Jan Vertonghen in spring 2013.
Key points
Micky Van de Ven nods into the net two times in first half
The home side's Jake O'Brien has goal disallowed after VAR review
Senegalese midfielder scores third in dying moments
The defeat is Everton's initial at Hill Dickinson Stadium
Micky van de Ven scored twice as Spurs prolonged their unbeaten run on the road in the top division and handed the Toffees their initial loss at Hill Dickinson Stadium.
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 end of the previous campaign.
But Tottenham's leader the Dutchman pounced on two occasions from set-pieces in the opening 45 minutes to put their impressive record at risk, and they were unable to turn it around before Pape Matar Sarr - set up by former Toffees attacker Richarlison - headed home a third to seal the victory.
Jake O'Brien thought he had equalized the scores five minutes after Van de Ven's first goal, glancing a bullet header past Guglielmo Vicario from an set-piece.
The Portuguese striker, meanwhile, failed to convert a number of opportunities, while Vicario made a series of fine saves to guarantee Tottenham went five away games without defeat in the Premier League.
But the video assistant referee stepped in, instructing Craig Pawson to check the score on the screen and the referee ruled that the winger and the attacker were in offside positions and had interfered with the Tottenham goalkeeper.
The win moves Thomas Frank's team into third place in the division, a handful of points off north London rivals and table-toppers the Gunners.
Everton stay in 14th position and sit half a dozen points ahead of the drop zone.
After a uninspiring midweek display in the Champions League against Monaco, where they managed a only attempts on goal, the manager chose Randal Kolo Muani in attack instead of ex-Everton forward the Brazilian.
The new arrival was failed to influence the game before being substituted for the substitute after an hour on the pitch, but it emerged he wasn't needed in the final outcome.
Centre-back the Dutchman converted into an attacking force to be feared.
Spurs' opening strike came from a cleverly executed corner routine, with the midfielder directing the ball toward the face of goal for an free the defender to head home.
And the captain - filling in as captain in Cristian Romero's unavailability - added his next goal in added time as he outjumped Jordan Pickford to nod in.
The Dutchman was equally outstanding in defence, partnering effectively with his defensive colleague as the pair dealt superbly with the 34 crosses sent their way by Everton.
Everton's recent stadium had been a fortress, but it appears the freshness may have faded somewhat and supporters headed to the exits after the midfielder's final goal for Spurs.
The end in Everton's unbeaten home run won't be a major worry for David Moyes, but their striker situation are.
Beto and Thierno Barry - the Everton's two established centre-forwards - had netted just a single goal combined before Sunday's outing.
And they were failed to add to that miserable record on a wet and gloomy evening on Merseyside.
Beto's form was epitomized when he chose to overhead kick a golden opportunity from near the goal, despite having space and time to reposition for a more conventional attempt that would have challenged Vicario.
Barry, as well, failed to make an impact after coming on for the starter in the 65th minute.
It wasn't for a lack of chance. The majority of Everton's offensive moves came through Grealish and the forward, who relentlessly delivered threatening balls into the Tottenham box.
Everton: Moyes' men head to Sunderland in the top flight on early November at 8 PM.
Spurs: Spurs travel to St James' Park on late October for their next round League Cup tie against the Magpies.
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