Micky van de Ven emerged as the initial backline player to net twice in a English top-flight match for Tottenham since Jan Vertonghen in March 2013.
Match summary
Micky Van de Ven nods home 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 the Merseyside club's first at Hill Dickinson Stadium
Micky van de Ven scored on two occasions as Spurs prolonged their unbeaten run on the road in the Premier League and delivered the Toffees their initial loss at their new home.
Everton were unbeaten in their last five matches at their recently opened stadium at Bramley-Moore Dock, having left their historic home behind after 133 years at the end of last season.
But Spurs captain Van de Ven pounced twice from corners in the first half to put their strong form at risk, and they were failed to turn it around before Pape Matar Sarr - assisted by ex-Everton forward Richarlison - headed home a third to put the game to bed.
Jake O'Brien believed he had equalized the scores a short while after Van de Ven's opener, directing a bullet header past the Spurs keeper from an Everton corner.
Beto, in the meantime, missed a number of chances, while Vicario made a succession of excellent saves to guarantee the London club went five away games unbeaten in the English top flight.
But the video assistant referee intervened, instructing Craig Pawson to review the goal on the screen and the referee ruled that Jack Grealish and Iliman Ndiaye were in an illegal position and had interfered with the Spurs keeper.
The win moves the manager's side into third place in the Premier League, a handful of points off local foes and leaders Arsenal.
Everton remain 14th and are six points clear of the drop zone.
Following a drab weekday display in the European competition against the French side, where they mustered a mere attempts on goal, the manager opted for Randal Kolo Muani in attack 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 pitch, but it turns out he proved unnecessary in the end.
Defender the Dutchman transformed into an offensive threat to be reckoned with.
Spurs' first goal came from a cleverly executed corner routine, with Rodrigo Bentancur heading the ball back across the six-yard box for an unmarked Van de Ven to score.
And Ven de Ven - filling in as skipper in the Argentine's absence - notched his next goal in added time as he outjumped the England keeper to nod in.
The Dutchman was equally impressive in defensive duties, combining well with his defensive colleague as the duo handled excellently with the numerous deliveries sent their way by the opposition.
The club's recent stadium had been a happy hunting ground, but it seems the freshness may have worn off slightly and fans flocked to the exits after the midfielder's third for Spurs.
The break in Everton's unbeaten home run will not be a massive worry for the manager, but their attacking woes are.
The forward and Thierno Barry - the Everton's pair of recognised centre-forwards - had netted just a single goal combined before the weekend's outing.
And they were unable to contribute to that poor tally on a damp and dreary evening on in Liverpool.
The striker's form was epitomized when he opted to overhead kick a clear chance from close range, even with having space and time to reposition for a simpler shot that would have challenged Vicario.
The substitute, as well, was unable to influence the game after replacing Beto in the second half.
It wasn't for a shortage of opportunity. The majority of Everton's offensive moves came through the winger and the forward, who tirelessly delivered threatening balls into the Tottenham penalty area.
Everton: Moyes' men head to Sunderland in the top flight on Monday, 3 November at 8 PM.
Tottenham: Spurs travel to Newcastle's ground on late October for their fourth-round Carabao Cup match against the Magpies.
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