Micky van de Ven emerged as the initial backline player to score two goals in a English top-flight game for Spurs since the Belgian defender in spring 2013.
Key points
Micky Van de Ven heads home twice 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 on two occasions as Tottenham extended their winning streak in away games in the Premier League and handed the Toffees their first defeat at their new home.
Everton were unbeaten in their last five matches at their recently opened stadium at Bramley-Moore Dock, having departed their historic home behind after 133 years at the end of last season.
But Tottenham's leader the Dutchman struck twice from set-pieces in the first half to put their impressive record at risk, and they were unable to mount a comeback before Pape Matar Sarr - set up by ex-Everton forward Richarlison - headed home a third to seal the victory.
The Everton defender thought he had equalized the tally a short while after Van de Ven's opener, glancing a powerful header past Guglielmo Vicario from an Everton corner.
The Portuguese striker, meanwhile, failed to convert a handful of opportunities, while Vicario made a succession of fine saves to ensure Tottenham went five away games unbeaten in the Premier League.
But the video assistant referee stepped in, instructing 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 Spurs keeper.
The win moves the manager's side into third place in the division, a handful of points off north London rivals and table-toppers the Gunners.
Everton stay 14th and are six points ahead of the relegation zone.
Following a drab midweek display in the European competition against Monaco, where they mustered a mere two shots on goal, Frank opted for the French striker in attack instead of ex-Everton forward Richarlison.
The summer signing was failed to influence the game before being substituted for Richarlison after an hour on the pitch, but it emerged he wasn't needed in the end.
Defender Van de Ven converted into an offensive threat to be feared.
Spurs' opening strike came from a well-worked set-piece play, with Rodrigo Bentancur heading the ball toward the six-yard box for an free Van de Ven to score.
And Ven de Ven - stepping up as skipper in Cristian Romero's absence - added his second in first-half stoppage time as he outjumped the England keeper to nod in.
The Dutchman was equally impressive in defence, partnering well with Kevin Danso as the duo dealt superbly with the 34 crosses thrown their way by Everton.
The club's new home had been a happy hunting ground, but it appears the novelty may have faded somewhat and fans headed to the exits after the midfielder's third for Tottenham.
The end in the team's unbeaten home run won't be a massive worry for David Moyes, but their striker situation are.
The forward and Thierno Barry - the Everton's two recognised strikers - had scored only one league goal between them before Sunday's match.
And they were unable to contribute to that miserable tally on a wet and gloomy evening on Merseyside.
The striker's performance was epitomized when he opted to bicycle-kick a clear chance from close range, despite having space and opportunity to reposition for a more conventional attempt that would have challenged the keeper.
The substitute, as well, was unable to make an impact after replacing the starter in the 65th minute.
It was not for a shortage of chance. The majority of the home side's offensive moves came through the winger and Ndiaye, who relentlessly delivered threatening crosses into the Spurs box.
The Toffees: Moyes' men head to Sunderland in the Premier League on Monday, 3 November at 20:00 GMT.
Tottenham: The London club go to St James' Park on Wednesday, 29 October for their next round League Cup match against Newcastle.
Lena is an environmental scientist and writer passionate about promoting sustainable practices through accessible, evidence-based content.