Micky van de Ven became the first backline player to net two goals in a Premier League match for Tottenham since Jan Vertonghen in spring 2013.
At a glance
Micky Van de Ven nods into the net two times in first half
Everton's Jake O'Brien has goal ruled after VAR review
Pape Matar Sarr nets third in dying moments
This loss is Everton's initial at Hill Dickinson Stadium
Micky van de Ven scored twice as Spurs extended their unbeaten run in away games in the top division and delivered the Toffees their first defeat at their new home.
Everton were unbeaten in their five previous fixtures at their new ground at Bramley-Moore Dock, having departed their historic home behind after more than a century at the conclusion of last season.
But Spurs captain the Dutchman struck twice from set-pieces in the opening 45 minutes to put their strong form at risk, and they were unable to turn it around before Pape Matar Sarr - assisted by former Toffees attacker the Brazilian - nodded in a final goal to put the game to bed.
Jake O'Brien believed he had levelled the tally five minutes after Van de Ven's opener, glancing a powerful header past the Spurs keeper from an set-piece.
The Portuguese striker, in the meantime, missed a number of chances, while Vicario made a succession of fine stops to ensure the London club went five away games unbeaten in the Premier League.
But the VAR intervened, instructing Craig Pawson to check the score on the screen and the referee decided that the winger and the attacker were in offside positions and had impeded with the Spurs keeper.
The win elevates the manager's team 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 relegation zone.
After a uninspiring midweek display in the European competition against the French side, where they mustered a mere attempts on goal, Frank chose the French striker in attack instead of former Toffees player Richarlison.
The summer signing was unable to influence the game before being replaced for the substitute after an hour on the field, but it emerged he wasn't needed in the end.
Centre-back Van de Ven converted into an attacking force to be reckoned with.
Tottenham's first goal came from a well-worked corner routine, with Rodrigo Bentancur directing the ball toward the face of goal for an free Van de Ven to score.
And the captain - stepping up as captain in Cristian Romero's absence - notched his next goal in first-half stoppage time as he outjumped the England keeper to head home.
Van de Ven was just as impressive in defence, partnering well with Kevin Danso as the pair dealt superbly with the numerous deliveries thrown their way by Everton.
The club's recent stadium had been a happy hunting ground, but it seems the freshness may have worn off slightly and fans headed to the exits after the midfielder's final goal for Tottenham.
The break in Everton's unbeaten home run will not be a massive worry for David Moyes, but their attacking woes are.
The forward and Thierno Barry - the Everton's pair of established centre-forwards - had netted just a single goal between them before Sunday's outing.
And they were failed to contribute to that poor tally on a damp and dreary evening on in Liverpool.
Beto's performance was epitomized when he opted to overhead kick a golden opportunity from near the goal, despite having room and time to reposition for a simpler attempt that would have challenged Vicario.
The substitute, as well, was unable to influence the game after coming on for the starter in the second half.
It wasn't for a shortage of chance. Most of Everton's attacks came through the winger and Ndiaye, who relentlessly whipped dangerous crosses into the Tottenham box.
Everton: Moyes' men travel to Sunderland in the top flight on Monday, 3 November at 8 PM.
Spurs: The London club travel to Newcastle's ground on late October for their next round Carabao Cup match against Newcastle.
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