Micky van de Ven became the initial defender to net two goals in a English top-flight game for Spurs since Jan Vertonghen in spring 2013.
At a glance
The Spurs defender nods home twice in first half
Everton's Jake O'Brien has equalizer ruled after VAR review
Senegalese midfielder scores third in 89th minute
This loss is the Merseyside club's initial at their new ground
The Tottenham defender scored on two occasions as Tottenham prolonged their winning streak on the road in the top division and delivered the Toffees their initial loss at Hill Dickinson Stadium.
The Toffees were undefeated 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 last season.
But Spurs captain Van de Ven struck twice from corners in the opening 45 minutes to place their impressive record at risk, and they were unable to turn it around before Pape Matar Sarr - set up by former Toffees attacker the Brazilian - headed home a third to put the game to bed.
Jake O'Brien believed he had equalized the scores five minutes after the defender's first goal, directing a powerful header past Guglielmo Vicario from an set-piece.
Beto, in the meantime, failed to convert a number of opportunities, while Vicario made a series of fine stops to guarantee the London club went five matches on the road without defeat in the Premier League.
But the VAR intervened, instructing Craig Pawson to check the score on the pitchside monitor and the referee decided that Jack Grealish and Iliman Ndiaye were in offside positions and had interfered with the Spurs keeper.
This victory elevates the manager's side up to third in the Premier League, a handful of points off local foes and leaders Arsenal.
The Merseyside club remain in 14th position and are six points ahead of the drop zone.
Following a drab weekday display in the Champions League against the French side, where they managed a only two shots on goal, Frank chose Randal Kolo Muani up front instead of ex-Everton forward the Brazilian.
The new arrival was unable to make much of an impact before being replaced for the substitute after 60 minutes on the field, but it turns out he wasn't needed in the final outcome.
Centre-back Van de Ven converted into an attacking force to be feared.
Spurs' opening strike came from a cleverly executed set-piece play, with Rodrigo Bentancur directing the ball back across the six-yard box for an free the defender to head home.
And the captain - stepping up as skipper in Cristian Romero's absence - notched his second in added time as he outjumped Jordan Pickford to head home.
The Dutchman was equally impressive in defence, combining well with his defensive colleague as the duo dealt superbly with the numerous deliveries thrown their way by the opposition.
Everton's recent stadium had been a fortress, but it seems the novelty may have faded somewhat and supporters flocked to the gates after Sarr's third for Spurs.
The break in Everton's winning streak at home won't be a major concern for the manager, but their striker situation will be.
Beto and Thierno Barry - the Everton's pair of established centre-forwards - had scored just one league goal between them before Sunday's match.
And they were failed to contribute to that poor record on a wet and gloomy evening on in Liverpool.
The striker's performance was epitomized when he opted to overhead kick a golden opportunity from close range, even with having space and opportunity to reposition for a more conventional shot that would have challenged Vicario.
The substitute, as well, was unable to influence the game after replacing the starter in the 65th minute.
It was not for a lack of chance. The majority of Everton's offensive moves came through Grealish and Ndiaye, who tirelessly delivered dangerous balls into the Tottenham box.
Everton: Moyes' men head to Sunderland in the top flight on Monday, 3 November at 20:00 GMT.
Spurs: Spurs travel to St James' Park on Wednesday, 29 October for their next round League Cup tie against the Magpies.
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