Micky van de Ven became the initial backline player to net two goals in a English top-flight game for Spurs since the Belgian defender in March 2013.
Match summary
The Spurs defender nods home twice in opening period
Everton's defender has equalizer disallowed after video check
Pape Matar Sarr nets third in dying moments
This loss is the Merseyside club's first at their new ground
The Tottenham defender found the net twice as Tottenham prolonged their unbeaten run in away games in the top division and handed the Toffees their initial loss at their new home.
The Toffees were unbeaten in their five previous fixtures at their recently opened stadium at Bramley-Moore Dock, having departed their historic home behind after more than a century at the conclusion of the previous campaign.
But Tottenham's leader the Dutchman pounced twice from corners in the first half to put their impressive record at jeopardy, and they were failed to turn it around before the midfielder - set up by former Toffees attacker Richarlison - headed home a final goal to seal the victory.
Jake O'Brien believed he had equalized the scores five minutes after Van de Ven's opener, directing a bullet header past Guglielmo Vicario from an set-piece.
Beto, meanwhile, failed to convert a handful of opportunities, while the Italian goalkeeper made a series of fine stops to guarantee Tottenham went five away games unbeaten in the Premier League.
But the video assistant referee intervened, advising Craig Pawson to check the goal on the screen and the referee decided that Jack Grealish and Iliman Ndiaye were in an illegal position and had impeded with the Spurs keeper.
The win elevates the manager's team up to third in the Premier League, five points off local foes and leaders Arsenal.
The Merseyside club stay 14th and sit six points clear of the drop zone.
Following a drab midweek display in the European competition against Monaco, where they managed a only attempts on target, Frank chose the French striker up front instead of former Toffees player the Brazilian.
The new arrival was failed to influence the game before being replaced for the substitute after an hour on the pitch, but it emerged he proved unnecessary in the final outcome.
Centre-back Van de Ven converted into an attacking force to be feared.
Spurs' opening strike came from a well-worked corner routine, with Rodrigo Bentancur directing the ball toward the face of goal for an unmarked the defender to head home.
And the captain - stepping up as skipper in Cristian Romero's unavailability - added his second in first-half stoppage time as he rose above Jordan Pickford to nod in.
Van de Ven was just as impressive in defensive duties, combining well with his defensive colleague as the pair dealt superbly with the numerous deliveries thrown their way by Everton.
Everton's new home had been a fortress, but it seems the novelty may have worn off somewhat and supporters flocked to the exits after Sarr's third for Tottenham.
The end in Everton's unbeaten home run will not be a massive worry for the manager, but their striker situation are.
Beto and the other striker - the Toffees' two recognised strikers - had scored only one league goal between them before the weekend's match.
And they were unable to contribute to that poor record on a damp and dreary evening on Merseyside.
The striker's performance was epitomized when he chose to bicycle-kick a golden opportunity from near the goal, even with having space and time to reposition for a more conventional shot that would have tested the keeper.
Barry, as well, was unable to influence the game after replacing Beto in the 65th minute.
It wasn't for a shortage of chance. Most of Everton's offensive moves came through the winger and the forward, who relentlessly whipped threatening balls into the Tottenham box.
The Toffees: Moyes' men travel to the Stadium of Light in the Premier League on Monday, 3 November at 8 PM.
Spurs: Spurs travel to Newcastle's ground on late October for their next round League Cup match against Newcastle.
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