Micky van de Ven became the initial defender to score twice in a Premier League match 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 goal ruled after video check
Pape Matar Sarr scores final goal in dying moments
This loss is the Merseyside club's initial at their new ground
The Tottenham defender found the net twice as Tottenham prolonged their unbeaten run in away games in the Premier League and delivered the Toffees their initial loss at Hill Dickinson Stadium.
Everton were undefeated in their last five matches at their recently opened stadium at the dockside location, having left Goodison Park behind after 133 years at the end of the previous campaign.
But Tottenham's leader Van de Ven pounced on two occasions from set-pieces in the first half to place their impressive record at jeopardy, and they were unable to turn it around before Pape Matar Sarr - 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 a short while after the defender's opener, glancing a powerful header past the Spurs keeper from an set-piece.
Beto, meanwhile, missed a number of opportunities, while Vicario made a series of fine saves to guarantee the London club went five matches on the road without defeat in the English top flight.
But the VAR intervened, advising Craig Pawson to check the goal on the screen and the referee decided that Jack Grealish and the attacker were in offside positions and had interfered with the Spurs keeper.
The win elevates Thomas Frank's team into third place in the division, five points off local foes and leaders Arsenal.
Everton remain 14th and are half a dozen points clear of the relegation zone.
Following a drab midweek display in the Champions League against Monaco, where they mustered a mere attempts on goal, the manager chose Randal Kolo Muani up front instead of ex-Everton forward Richarlison.
The new arrival was failed to make much of an impact before being substituted for the substitute after an hour on the pitch, but it emerged he wasn't needed in the end.
Defender Van de Ven converted into an offensive threat to be reckoned with.
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 the defender to score.
And Ven de Ven - stepping up as skipper in Cristian Romero's unavailability - notched his second in first-half stoppage time as he outjumped Jordan Pickford to head home.
Van de Ven was just as outstanding in defensive duties, partnering effectively with his defensive colleague as the duo handled excellently with the numerous deliveries sent their way by Everton.
The club's recent stadium had been a happy hunting ground, but it seems the novelty may have faded slightly and fans headed to the exits after the midfielder's final goal for Spurs.
The break in the team's unbeaten home run won't be a massive concern for the manager, but their attacking woes will be.
Beto and the other striker - the Toffees' two established strikers - had netted only a single goal between them before the weekend's match.
And they were failed to contribute to that poor tally on a wet and gloomy evening on Merseyside.
The striker's performance was epitomized when he opted to overhead kick a clear chance from near the goal, even with having room and opportunity to readjust for a simpler attempt that would have challenged Vicario.
Barry, too, was unable to influence the game after coming on for the starter in the second half.
It was not for a lack of opportunity. Most of the home side's attacks came through Grealish and Ndiaye, who relentlessly delivered threatening crosses into the Tottenham box.
The Toffees: The team head to Sunderland in the Premier League on early November at 8 PM.
Spurs: The London club go to St James' Park on Wednesday, 29 October for their fourth-round Carabao Cup match against the Magpies.
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