Micky van de Ven emerged as the first backline player to net two goals in a Premier League match for Spurs since Jan Vertonghen in March 2013.
At a glance
The Spurs defender heads into the net twice in opening period
Everton's defender has equalizer disallowed after video check
Pape Matar Sarr scores third in 89th minute
This loss is Everton's initial at their new ground
Micky van de Ven scored twice as Spurs extended their winning streak in away games in the top division and delivered Everton their initial loss at Hill Dickinson Stadium.
Everton were undefeated in their five previous fixtures at their new ground at Bramley-Moore Dock, having left their historic home behind after 133 years at the end of last season.
But Spurs captain the Dutchman struck twice from set-pieces in the first half to place their impressive record at risk, and they were unable to mount a comeback before the midfielder - assisted by ex-Everton forward Richarlison - nodded in a final goal to seal the victory.
The Everton defender believed he had equalized the tally a short while after Van de Ven's first goal, directing a powerful header past the Spurs keeper from an set-piece.
Beto, meanwhile, missed a handful of chances, while the Italian goalkeeper made a series of excellent saves to guarantee Tottenham went five matches on the road unbeaten in the English top flight.
But the video assistant referee stepped in, advising the match official to check the score on the pitchside monitor and the referee decided that Jack Grealish and the attacker were in offside positions and had interfered with the Spurs keeper.
The win elevates Thomas Frank's team up to third in the Premier League, a handful of points off north London rivals and leaders the Gunners.
Everton remain 14th and are six points clear of the relegation zone.
After a uninspiring midweek performance in the Champions League against the French side, where they managed a only two shots on target, Frank chose Randal Kolo Muani in attack instead of former Toffees player the Brazilian.
The new arrival was unable to influence the game before being replaced for Richarlison after 60 minutes on the field, but it emerged he wasn't needed in the end.
Centre-back the Dutchman converted into an attacking force to be reckoned with.
Spurs' opening strike came from a well-worked set-piece play, with Rodrigo Bentancur directing 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 Cristian Romero's absence - added his second in first-half stoppage time as he rose above Jordan Pickford to head home.
The Dutchman was just as outstanding in defensive duties, combining well with his defensive colleague as the pair dealt superbly with the 34 crosses thrown their way by Everton.
The club's new home had been a happy hunting ground, but it seems the freshness may have faded somewhat and supporters flocked to the gates after Sarr's final goal for Tottenham.
The end in Everton's winning streak at home will not be a major concern for the manager, but their attacking woes are.
Beto and Thierno Barry - the Toffees' pair of established centre-forwards - had scored just one league goal between them before Sunday's match.
And they were unable to contribute to that poor tally on a wet and gloomy night on Merseyside.
The striker's form was epitomized when he chose to overhead kick a golden opportunity from near the goal, even with having space and opportunity to reposition for a more conventional shot that would have challenged Vicario.
Barry, too, was unable to make an impact after replacing Beto in the 65th minute.
It wasn't for a shortage of opportunity. The majority of Everton's offensive moves came through Grealish and the forward, who tirelessly whipped threatening balls into the Tottenham penalty area.
Everton: Moyes' men travel to Sunderland in the top flight on early November at 20:00 GMT.
Tottenham: The London club go to Newcastle's ground on Wednesday, 29 October for their next round League Cup match against Newcastle.
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