Micky van de Ven became the initial backline player to score two goals in a Premier League game for Spurs since the Belgian defender in spring 2013.
Match summary
The Spurs defender heads home two times in opening period
The home side's Jake O'Brien has goal disallowed after video check
Senegalese midfielder nets third in 89th minute
The defeat is Everton's first at their new ground
The Tottenham defender scored twice as Spurs prolonged their unbeaten run in away games in the Premier League and delivered Everton their initial loss at Hill Dickinson Stadium.
The Toffees were undefeated in their five previous fixtures at their recently opened stadium at the dockside location, having departed their historic home behind after 133 years at the end of the previous campaign.
But Spurs captain the Dutchman struck twice from corners in the opening 45 minutes to put their impressive record at risk, and they were unable to mount a comeback before Pape Matar Sarr - set up by former Toffees attacker Richarlison - headed home a final goal to seal the victory.
The Everton defender thought he had equalized the tally five minutes after Van de Ven's first goal, glancing a bullet header past the Spurs keeper from an set-piece.
The Portuguese striker, in the meantime, failed to convert a handful of opportunities, while the Italian goalkeeper made a series of excellent saves to guarantee Tottenham went five away games unbeaten in the Premier League.
But the video assistant referee intervened, advising Craig Pawson to review the score on the screen and the referee ruled that the winger and Iliman Ndiaye were in an illegal position and had interfered with the Tottenham goalkeeper.
This victory elevates the manager's team up to third in the Premier League, a handful of points off north London rivals and table-toppers the Gunners.
The Merseyside club remain in 14th position and are six points clear of the relegation zone.
Following a drab weekday performance in the Champions League against Monaco, where they managed a only two shots on target, Frank chose the French striker in attack instead of ex-Everton forward the Brazilian.
The new arrival was unable to make much of an impact before being substituted for Richarlison after 60 minutes on the pitch, but it turns out he wasn't needed in the end.
Centre-back the Dutchman transformed into an attacking force to be feared.
Tottenham's opening strike came from a well-worked corner routine, with Rodrigo Bentancur directing the ball back across the face of goal for an free Van de Ven to head home.
And the captain - stepping up as skipper in Cristian Romero's absence - added his second in first-half stoppage time as he rose above the England keeper to head home.
The Dutchman was just as outstanding in defensive duties, combining well with his defensive colleague as the pair dealt excellently with the 34 crosses sent their way by Everton.
The club's recent stadium had been a fortress, but it appears the freshness may have faded slightly and supporters flocked to the exits after Sarr's third for Tottenham.
The end in Everton's unbeaten home run will not be a major concern for the manager, but their striker situation will be.
The forward and the other striker - the Toffees' two established strikers - had scored just a single goal combined before the weekend's match.
And they were unable to contribute to that poor record on a wet and gloomy evening on Merseyside.
Beto's performance was summed up when he chose to overhead kick a clear chance from close range, even with having room and time to readjust for a more conventional attempt that would have tested Vicario.
The substitute, too, was unable to make an impact after coming on for Beto in the second half.
It was not for a shortage of chance. Most of Everton's attacks came through the winger and the forward, who relentlessly delivered dangerous balls into the Spurs penalty area.
The Toffees: Moyes' men head to Sunderland in the Premier League on early November at 20:00 GMT.
Tottenham: Spurs go to Newcastle's ground on late October for their fourth-round League Cup match against the Magpies.
A tech enthusiast and journalist with over a decade of experience covering emerging technologies and digital transformations.
Michael Hunter
Michael Hunter
Michael Hunter
Michael Hunter
Michael Hunter
Michael Hunter