Micky van de Ven became the first defender to score twice in a English top-flight match for Spurs since the Belgian defender in March 2013.
Key points
The Spurs defender heads home two times in first half
The home side's defender has equalizer ruled after video check
Pape Matar Sarr scores final goal in 89th minute
This loss is Everton's initial at Hill Dickinson Stadium
Micky van de Ven scored twice as Spurs prolonged their unbeaten run in away games in the Premier League and delivered Everton their initial loss at their new home.
The Toffees were undefeated in their five previous fixtures at their recently opened stadium at the dockside location, having left Goodison Park behind after 133 years at the end of the previous campaign.
But Spurs captain the Dutchman struck on two occasions from corners in the first half to place their impressive record at jeopardy, and they were unable to mount a comeback before Pape Matar Sarr - set up by ex-Everton forward the Brazilian - headed home a final goal to seal the victory.
The Everton defender believed he had equalized the scores five minutes after the defender's first goal, glancing a bullet header past Guglielmo Vicario from an Everton corner.
Beto, in the meantime, missed a handful of chances, while the Italian goalkeeper made a succession of excellent stops to guarantee Tottenham went five matches on the road without defeat in the English top flight.
But the VAR intervened, advising Craig Pawson to check the score on the screen and the referee ruled that Jack Grealish and the attacker were in an illegal position and had interfered with the Tottenham goalkeeper.
The win elevates the manager's team into third place in the Premier League, five points off local foes and leaders the Gunners.
Everton stay 14th and sit six points clear of the drop zone.
After a uninspiring weekday performance in the Champions League against the French side, where they managed a only two shots on target, Frank opted for Randal Kolo Muani in attack instead of former Toffees player the Brazilian.
The new arrival was unable to make much of an impact before being substituted for the substitute after an hour on the field, but it turns out he wasn't needed in the final outcome.
Defender the Dutchman transformed into an attacking force to be feared.
Spurs' first goal came from a cleverly executed set-piece play, with Rodrigo Bentancur heading the ball back across the six-yard box for an unmarked Van de Ven to head home.
And Ven de Ven - filling in as skipper in Cristian Romero's absence - added his second in first-half stoppage time as he outjumped Jordan Pickford to nod in.
Van de Ven was just as impressive in defensive duties, combining effectively with his defensive colleague as the duo dealt superbly with the numerous deliveries thrown their way by Everton.
Everton's new home had been a happy hunting ground, but it appears the freshness may have faded slightly and fans headed to the gates after the midfielder's third for Tottenham.
The break in Everton's unbeaten home run won't be a major worry for the manager, but their attacking woes will be.
The forward and Thierno Barry - the Toffees' pair of established strikers - had netted only a single goal combined before the weekend's outing.
And they were failed to contribute to that poor record on a damp and dreary night on in Liverpool.
The striker's performance was summed up when he opted to bicycle-kick a golden opportunity from near the goal, despite having room and time to readjust for a more conventional shot that would have challenged Vicario.
Barry, as well, was unable to make an impact after replacing Beto in the second half.
It was not for a shortage of chance. The majority of Everton's attacks came through Grealish and the forward, who relentlessly delivered threatening crosses into the Tottenham box.
Everton: The team head to Sunderland in the top flight on early November at 20:00 GMT.
Spurs: Spurs travel to Newcastle's ground on late October for their next round Carabao Cup tie against the Magpies.
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