Micky van de Ven emerged as the first backline player to score two goals in a Premier League match for Spurs since the Belgian defender in March 2013.
At a glance
The Spurs defender heads home twice in first half
Everton's defender has equalizer ruled after video check
Pape Matar Sarr scores final goal in 89th minute
This loss is Everton's initial at their new ground
The Tottenham defender scored twice as Tottenham prolonged their winning streak in away games in the Premier League and handed Everton their first defeat at Hill Dickinson Stadium.
The Toffees were unbeaten in their last five matches at their new ground at the dockside location, having left Goodison Park behind after more than a century at the end of the previous campaign.
But Spurs captain the Dutchman struck twice from corners in the first half to place their strong form at jeopardy, and they were unable to mount a comeback before Pape Matar Sarr - set up by former Toffees attacker the Brazilian - headed home a final goal to put the game to bed.
Jake O'Brien believed he had levelled the tally a short while after Van de Ven's opener, glancing a bullet header past Guglielmo Vicario from an set-piece.
Beto, meanwhile, failed to convert a handful of opportunities, while Vicario made a series of excellent saves to ensure the London club went five matches on the road unbeaten in the Premier League.
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 Tottenham goalkeeper.
This victory elevates Thomas Frank's team into third place in the division, a handful of points off north London rivals and leaders Arsenal.
The Merseyside club remain 14th and are six points clear of the relegation zone.
Following a uninspiring midweek display in the Champions League against Monaco, where they mustered a mere attempts on goal, the manager opted for Randal Kolo Muani in attack instead of former Toffees player Richarlison.
The summer signing was unable to make much of an impact before being replaced for Richarlison after 60 minutes on the pitch, but it turns out he wasn't needed in the final outcome.
Centre-back Van de Ven transformed into an attacking force to be feared.
Tottenham's opening strike came from a cleverly executed set-piece play, with Rodrigo Bentancur directing the ball toward the six-yard box for an unmarked Van de Ven to head home.
And the captain - stepping up as skipper in the Argentine's absence - notched his next goal in added time as he rose above the England keeper to nod in.
Van de Ven was just as impressive in defence, partnering effectively with his defensive colleague as the pair handled superbly with the numerous deliveries thrown their way by the opposition.
Everton's recent stadium had been a fortress, but it appears the novelty may have faded slightly and supporters headed to the exits after Sarr's third for Tottenham.
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 Everton's pair of recognised centre-forwards - had scored only a single goal combined before the weekend's match.
And they were unable to contribute to that poor record on a damp and dreary night on in Liverpool.
Beto's performance was epitomized when he opted to overhead kick a golden opportunity from close range, even with having room and time to reposition for a simpler attempt that would have challenged the keeper.
The substitute, as well, was unable to make an impact after replacing the starter in the second half.
It was not for a shortage of opportunity. Most of Everton's attacks came through Grealish and Ndiaye, who tirelessly delivered dangerous crosses into the Tottenham box.
Everton: Moyes' men travel to the Stadium of Light in the Premier League on Monday, 3 November at 20:00 GMT.
Spurs: The London club go to Newcastle's ground on Wednesday, 29 October for their next round Carabao Cup match against the Magpies.