Micky van de Ven emerged as the first backline player to score twice in a English top-flight game for Spurs since Jan Vertonghen in March 2013.
At a glance
The Spurs defender heads into the net two times in opening period
Everton's defender has goal disallowed after VAR review
Senegalese midfielder nets third in 89th minute
The defeat is Everton's first at their new ground
Micky van de Ven found the net on two occasions as Tottenham prolonged their unbeaten run in away games in the Premier League and delivered the Toffees their first defeat at Hill Dickinson Stadium.
Everton were undefeated in their last five matches at their new ground at the dockside location, having departed their historic home behind after 133 years at the conclusion of last season.
But Spurs captain the Dutchman pounced twice from corners in the first half to put their strong form at jeopardy, and they were unable to mount a comeback before Pape Matar Sarr - assisted by former Toffees attacker Richarlison - nodded in a final goal to put the game to bed.
Jake O'Brien believed he had levelled the scores a short while after the defender's first goal, glancing a bullet header past Guglielmo Vicario from an Everton corner.
Beto, in the meantime, failed to convert a handful of opportunities, while Vicario made a succession of fine saves to ensure the London club went five away games without defeat in the Premier League.
But the VAR intervened, advising the match official to review the goal on the pitchside monitor and the official decided that Jack Grealish and the attacker were in offside positions and had impeded with the Spurs keeper.
The win elevates the manager's side into third place in the Premier League, a handful of points off north London rivals and leaders the Gunners.
The Merseyside club remain 14th and are half a dozen points clear of the drop zone.
Following a uninspiring midweek performance in the European competition against the French side, where they managed a only attempts on target, the manager chose the French striker up front instead of ex-Everton forward Richarlison.
The new arrival was failed to influence the game before being substituted for Richarlison after an hour on the field, but it emerged he wasn't needed in the end.
Centre-back the Dutchman converted into an offensive threat 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 the defender to head home.
And Ven de Ven - filling in as captain in the Argentine's unavailability - added his next goal in first-half stoppage time as he rose above the England keeper to head home.
The Dutchman was equally outstanding in defence, combining effectively with Kevin Danso as the duo handled excellently with the 34 crosses thrown their way by Everton.
The club's recent stadium had been a fortress, but it seems the freshness may have worn off somewhat and fans headed to the exits after Sarr's third for Spurs.
The break in Everton's winning streak at home will not be a major concern for the manager, but their attacking woes are.
Beto and the other striker - the Everton's two established strikers - had scored just a single goal between them before the weekend's outing.
And they were failed to add to that poor tally on a damp and dreary evening on in Liverpool.
The striker's performance was summed up when he opted to bicycle-kick a clear chance from near the goal, even with having space and time to readjust for a simpler shot that would have tested Vicario.
Barry, as well, failed to influence the game after replacing Beto in the 65th minute.
It wasn't for a shortage of chance. The majority of the home side's offensive moves came through Grealish and Ndiaye, who tirelessly whipped dangerous crosses into the Tottenham penalty area.
Everton: Moyes' men travel to the Stadium of Light in the Premier League on Monday, 3 November at 20:00 GMT.
Spurs: Spurs go to Newcastle's ground on late October for their next round Carabao Cup match against Newcastle.