Micky van de Ven became the initial backline player to net two goals in a Premier League match for Tottenham since the Belgian defender in March 2013.
Match summary
Micky Van de Ven heads into the net twice in opening period
Everton's defender has equalizer ruled after VAR review
Pape Matar Sarr nets final goal in dying moments
The defeat is Everton's initial at their new ground
The Tottenham defender found the net twice as Spurs extended their winning streak on the road in the top division and delivered the Toffees their first defeat at Hill Dickinson Stadium.
Everton were unbeaten in their last five matches at their recently opened stadium at Bramley-Moore Dock, having departed their historic home behind after 133 years at the end of last season.
But Spurs captain Van de Ven pounced on two occasions 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 - nodded in a third to put the game to bed.
The Everton defender believed he had levelled the scores a short while after the defender's first goal, directing a powerful header past the Spurs keeper from an Everton corner.
The Portuguese striker, meanwhile, failed to convert a number of chances, while the Italian goalkeeper made a series of excellent saves to ensure the London club went five matches on the road without defeat in the English top flight.
But the VAR intervened, advising Craig Pawson to review the goal on the screen and the official ruled that Jack Grealish and the attacker were in offside positions and had impeded with the Spurs keeper.
The win moves the manager's side up to third in the division, five points off north London rivals and table-toppers the Gunners.
The Merseyside club stay 14th and are six points clear of the relegation zone.
After a uninspiring weekday display in the Champions League against the French side, where they managed a mere attempts on goal, the manager chose the French striker up front instead of ex-Everton forward the Brazilian.
The summer signing was unable to influence the game before being substituted for Richarlison after 60 minutes on the field, but it turns out 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 set-piece play, with the midfielder heading the ball back across the six-yard box for an unmarked Van de Ven to score.
And Ven de Ven - stepping up as captain in the Argentine's unavailability - added his next goal in added time as he rose above the England keeper to head home.
The Dutchman was equally impressive in defence, partnering well with his defensive colleague as the duo handled superbly with the 34 crosses sent their way by Everton.
Everton's new home had been a happy hunting ground, but it appears the freshness may have faded slightly and supporters flocked to the gates after the midfielder's third for Tottenham.
The break in the team's unbeaten home run won't be a massive concern for David Moyes, but their striker situation will be.
Beto and the other striker - the Toffees' pair of recognised centre-forwards - had scored just a single goal between them before Sunday's outing.
And they were unable to add to that miserable record on a damp and dreary evening on in Liverpool.
Beto's performance 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 shot that would have tested the keeper.
Barry, as well, was unable to make an impact after coming on for Beto in the second half.
It wasn't for a lack of opportunity. The majority of the home side's attacks came through Grealish and Ndiaye, who relentlessly whipped threatening crosses into the Tottenham penalty area.
Everton: The team travel to Sunderland in the top flight on early November at 8 PM.
Tottenham: Spurs travel to St James' Park on Wednesday, 29 October for their next round Carabao Cup tie against the Magpies.