Micky van de Ven emerged as the initial defender to score twice in a English top-flight match for Tottenham since Jan Vertonghen in spring 2013.
Key points
Micky Van de Ven nods into the net two times in first half
The home side's defender has goal disallowed after video check
Senegalese midfielder nets third in dying moments
The defeat is the Merseyside club's initial at Hill Dickinson Stadium
The Tottenham defender scored twice as Tottenham extended their winning streak on the road in the top division and delivered Everton their first defeat at their new home.
The Toffees were unbeaten in their last five matches at their recently opened stadium at the dockside location, having departed Goodison Park behind after more than a century at the conclusion of the previous campaign.
But Tottenham's leader the Dutchman pounced on two occasions from corners in the first half to put their impressive record at jeopardy, and they were unable to mount a comeback before Pape Matar Sarr - assisted by ex-Everton forward Richarlison - headed home a third to seal the victory.
The Everton defender thought he had equalized the tally five minutes after Van de Ven's opener, glancing a bullet header past Guglielmo Vicario from an Everton corner.
The Portuguese striker, in the meantime, failed to convert a number of chances, while the Italian goalkeeper made a series of fine stops to ensure Tottenham went five matches on the road without defeat in the English top flight.
But the VAR stepped in, advising the match official to review the score on the screen and the official decided that Jack Grealish and Iliman Ndiaye were in offside positions and had interfered with the Spurs keeper.
The win elevates the manager's side into third place in the division, five points off north London rivals and table-toppers the Gunners.
The Merseyside club remain in 14th position and are six points clear of the drop zone.
Following a drab midweek performance in the Champions League against the French side, where they mustered a mere attempts on goal, Frank opted for the French striker up front instead of former Toffees player Richarlison.
The new arrival was failed to influence the game before being replaced for Richarlison after an hour on the pitch, but it emerged he wasn't needed in the final outcome.
Defender the Dutchman transformed into an attacking force to be reckoned with.
Tottenham's first goal came from a well-worked set-piece play, with the midfielder directing the ball back across the six-yard box for an unmarked the defender to score.
And the captain - filling in as skipper in the Argentine's absence - notched 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 defence, combining effectively with his defensive colleague as the duo dealt excellently with the 34 crosses thrown their way by the opposition.
Everton's new home had been a fortress, but it seems the novelty may have faded somewhat and fans flocked to the exits after Sarr's final goal for Spurs.
The break in Everton's unbeaten home run will not be a major concern for the manager, but their striker situation will be.
Beto and Thierno Barry - the Toffees' two established strikers - had scored just one league goal combined before the weekend's outing.
And they were failed to add to that miserable tally on a damp and dreary evening on in Liverpool.
Beto's performance was epitomized when he opted to bicycle-kick a golden opportunity from near the goal, despite having space and opportunity to reposition for a more conventional shot that would have tested Vicario.
Barry, as well, was unable to make an impact after coming on for the starter in the 65th minute.
It was not for a lack of chance. Most of Everton's offensive moves came through Grealish and the forward, who relentlessly whipped threatening balls into the Tottenham box.
Everton: Moyes' men head to the Stadium of Light in the Premier League on Monday, 3 November at 20:00 GMT.
Tottenham: Spurs go to Newcastle's ground on late October for their next round League Cup tie against Newcastle.