Micky van de Ven emerged as the first backline player to score twice in a Premier League match for Tottenham since Jan Vertonghen in spring 2013.
Match summary
Micky Van de Ven heads home two times in opening period
The home side's Jake O'Brien has goal disallowed after video check
Senegalese midfielder nets final goal in 89th minute
This loss is Everton's first at their new ground
Micky van de Ven found the net twice as Tottenham extended their winning streak on the road in the Premier League and delivered the Toffees their initial loss at their new home.
Everton were undefeated in their last five matches at their recently opened stadium at the dockside location, having left Goodison Park behind after more than a century at the conclusion of last season.
But Tottenham's leader the Dutchman struck twice from set-pieces in the opening 45 minutes to place their impressive record at risk, and they were unable to turn it around before the midfielder - assisted by ex-Everton forward the Brazilian - nodded in a final goal to put the game to bed.
The Everton defender believed he had equalized the tally five minutes after Van de Ven's first goal, directing a powerful header past the Spurs keeper from an Everton corner.
Beto, in the meantime, missed a handful of opportunities, while Vicario made a series of fine stops to guarantee the London club went five away games without defeat in the Premier League.
But the video assistant referee stepped in, advising the match official to check the goal on the pitchside monitor and the referee decided that Jack Grealish and the attacker were in offside positions and had interfered with the Tottenham goalkeeper.
The win elevates the manager's team into third place in the division, five points off north London rivals and leaders Arsenal.
The Merseyside club stay in 14th position and sit six points ahead of the drop zone.
After a uninspiring midweek performance in the Champions League against the French side, where they managed a only attempts on target, Frank chose the French striker in attack instead of ex-Everton forward Richarlison.
The new arrival was failed to influence the game before being replaced for Richarlison after 60 minutes on the pitch, but it emerged he proved unnecessary in the end.
Centre-back the Dutchman converted into an offensive threat to be reckoned with.
Spurs' first goal came from a well-worked set-piece play, with the midfielder heading the ball toward the six-yard box for an unmarked the defender to head home.
And the captain - stepping up as skipper in the Argentine's absence - added his next goal in first-half stoppage time as he rose above Jordan Pickford to head home.
Van de Ven was just as impressive in defensive duties, partnering well with Kevin Danso as the pair dealt superbly with the numerous deliveries thrown their way by the opposition.
The club's new home had been a happy hunting ground, but it seems the novelty may have faded slightly and fans headed to the exits after Sarr's third for Spurs.
The break in Everton's unbeaten home run won't be a major worry for the manager, but their attacking woes are.
Beto and the other striker - the Toffees' two recognised strikers - had netted only one league goal combined before the weekend's outing.
And they were failed to add to that poor record on a damp and dreary evening on Merseyside.
The striker's form was summed up when he opted to overhead kick a clear chance from near the goal, even with having space and time to reposition for a simpler shot that would have tested Vicario.
The substitute, too, failed to influence the game after replacing the starter in the 65th minute.
It was not for a lack of opportunity. The majority of Everton's offensive moves came through Grealish and Ndiaye, who tirelessly whipped dangerous crosses into the Tottenham box.
Everton: The team travel to the Stadium of Light in the top flight on early November at 8 PM.
Tottenham: The London club travel to St James' Park on Wednesday, 29 October for their fourth-round Carabao Cup tie against Newcastle.
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