Micky van de Ven became the initial backline player to net two goals in a Premier League game for Spurs since the Belgian defender in March 2013.
At a glance
The Spurs defender heads into the net twice in opening period
The home side's defender has equalizer ruled after VAR review
Pape Matar Sarr nets third in 89th minute
The defeat is the Merseyside club's first at their new ground
The Tottenham defender scored on two occasions as Tottenham extended their winning streak in away games in the Premier League and handed the Toffees their first defeat at Hill Dickinson Stadium.
The Toffees were unbeaten in their five previous fixtures at their new ground at the dockside location, having departed Goodison Park behind after 133 years at the conclusion of last season.
But Spurs captain Van de Ven struck on two occasions from corners in the first half to put their strong form at jeopardy, and they were unable to mount a comeback before the midfielder - set up by former Toffees attacker the Brazilian - headed home a third to seal the victory.
The Everton defender believed he had levelled the scores five minutes after Van de Ven's first goal, directing a bullet header past the Spurs keeper from an Everton corner.
Beto, meanwhile, missed a handful of chances, while the Italian goalkeeper made a succession of fine saves to ensure Tottenham went five matches on the road without defeat in the English top flight.
But the VAR stepped in, advising the match official to check the goal on the pitchside monitor and the referee decided that Jack Grealish and the attacker were in offside positions and had interfered with the Spurs keeper.
The win moves the manager's side into third place in the division, five points off north London rivals and table-toppers the Gunners.
Everton stay 14th and sit six points ahead of the relegation zone.
Following a drab midweek display in the Champions League against Monaco, where they managed a only attempts on target, the manager chose Randal Kolo Muani up front instead of ex-Everton forward the Brazilian.
The summer signing was failed to influence the game before being substituted for Richarlison after an hour on the pitch, but it turns out he proved unnecessary in the end.
Defender Van de Ven transformed into an attacking force to be feared.
Tottenham's opening strike came from a well-worked corner routine, with Rodrigo Bentancur heading the ball toward the six-yard box for an free Van de Ven to score.
And the captain - filling in as skipper in the Argentine's unavailability - notched his next goal in added time as he rose above the England keeper to head home.
Van de Ven was equally outstanding in defence, combining effectively with his defensive colleague as the duo dealt excellently with the 34 crosses thrown their way by Everton.
The club's recent stadium had been a happy hunting ground, but it seems the freshness may have worn off somewhat and fans flocked to the exits after the midfielder's final goal for Tottenham.
The break in the team's unbeaten home run will not be a major concern for the manager, but their attacking woes will be.
The forward and the other striker - the Everton's pair of established centre-forwards - had scored just a single goal combined before Sunday's outing.
And they were unable to contribute to that poor tally on a wet and gloomy evening on Merseyside.
The striker's form was epitomized when he chose to bicycle-kick a golden opportunity from near the goal, despite having room and time to reposition for a simpler shot that would have tested Vicario.
The substitute, as well, failed to make an impact after replacing the starter in the 65th minute.
It wasn't for a shortage of chance. The majority of Everton's attacks came through the winger and Ndiaye, who tirelessly whipped threatening crosses into the Spurs penalty area.
Everton: The team head to the Stadium of Light in the top flight on early November at 20:00 GMT.
Tottenham: Spurs travel to St James' Park on late October for their fourth-round Carabao Cup match against Newcastle.
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