Micky van de Ven became the initial defender to score two goals in a Premier League match for Spurs since the Belgian defender in March 2013.
At a glance
Micky Van de Ven nods home twice in opening period
Everton's defender has goal ruled after VAR review
Pape Matar Sarr scores final goal in 89th minute
The defeat is Everton's first at their new ground
Micky van de Ven found the net on two occasions as Spurs prolonged their unbeaten run on the road in the top division and delivered Everton their first defeat at their new home.
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 Tottenham's leader the Dutchman pounced on two occasions from set-pieces in the opening 45 minutes to place their impressive record at risk, and they were unable to turn it around before the midfielder - assisted by former Toffees attacker the Brazilian - nodded in a third to put the game to bed.
Jake O'Brien thought he had levelled the tally a short while after Van de Ven's first goal, glancing a powerful header past Guglielmo Vicario from an set-piece.
Beto, in the meantime, missed a handful of chances, while the Italian goalkeeper made a series of excellent stops to ensure the London club went five away games unbeaten in the English top flight.
But the VAR stepped in, instructing Craig Pawson to check the score on the pitchside monitor and the official ruled that Jack Grealish and Iliman Ndiaye were in an illegal position and had impeded with the Tottenham goalkeeper.
The win elevates Thomas Frank's side into third place in the Premier League, five points off local foes and leaders the Gunners.
Everton remain 14th and sit half a dozen points clear of the drop zone.
Following a uninspiring weekday performance in the Champions League against the French side, where they managed a only two shots on target, the manager chose Randal Kolo Muani in attack instead of ex-Everton forward the Brazilian.
The summer signing was failed to make much of an impact before being replaced for the substitute after an hour on the pitch, but it turns out he proved unnecessary in the end.
Defender Van de Ven transformed into an offensive threat to be reckoned with.
Spurs' first goal came from a cleverly executed set-piece play, with Rodrigo Bentancur heading the ball back across the six-yard box for an unmarked Van de Ven to head home.
And Ven de Ven - filling in as skipper in Cristian Romero's absence - notched his second in added time as he outjumped the England keeper to nod in.
The Dutchman was just as outstanding in defence, combining well with Kevin Danso as the duo handled excellently with the numerous deliveries sent their way by the opposition.
Everton's recent stadium had been a fortress, but it appears the novelty may have worn off slightly and fans headed to the gates after Sarr's third for Tottenham.
The break in the team's winning streak at home will not be a massive concern for David Moyes, but their striker situation will be.
The forward and the other striker - the Everton's pair of established strikers - had scored just a single goal between them before the weekend's outing.
And they were failed to add to that miserable record on a damp and dreary evening on Merseyside.
The striker's performance was summed up when he chose to overhead kick a clear chance from close range, despite having space and time to readjust for a more conventional attempt that would have challenged Vicario.
The substitute, too, failed to make an impact after replacing Beto in the 65th minute.
It wasn't for a shortage of chance. The majority of Everton's attacks came through the winger and the forward, who tirelessly whipped dangerous crosses into the Tottenham box.
The Toffees: The team travel to Sunderland in the top flight on early November at 8 PM.
Tottenham: The London club travel to St James' Park on late October for their next round League Cup tie against the Magpies.