Micky van de Ven emerged as the initial defender to score two goals in a English top-flight match for Spurs since the Belgian defender in spring 2013.
Key points
Micky Van de Ven heads into the net two times in first half
Everton's Jake O'Brien has equalizer ruled after VAR review
Senegalese midfielder scores final goal in 89th minute
This loss is the Merseyside club's first at their new ground
Micky van de Ven found the net twice as Spurs prolonged their unbeaten run on the road in the top division 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 more than a century at the end of last season.
But Spurs captain Van de Ven struck on two occasions from corners in the first half to place their strong form at jeopardy, and they were unable to mount a comeback before Pape Matar Sarr - assisted by former Toffees attacker the Brazilian - nodded in a final goal to seal the victory.
The Everton defender believed he had levelled the tally a short while after Van de Ven's first goal, directing a powerful header past Guglielmo Vicario from an set-piece.
The Portuguese striker, meanwhile, failed to convert a number of opportunities, while Vicario made a series of excellent saves to ensure the London club went five matches on the road unbeaten in the Premier League.
But the VAR stepped in, instructing the match official to review the goal on the pitchside monitor and the official ruled that the winger and the attacker were in offside positions and had interfered with the Spurs keeper.
The win elevates the manager's side up to third in the Premier League, a handful of points off north London rivals and table-toppers Arsenal.
Everton remain in 14th position and sit half a dozen points clear of the drop zone.
Following a drab weekday performance in the Champions League against Monaco, where they managed a only attempts on target, Frank opted for Randal Kolo Muani up front 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 turns out he proved unnecessary in the end.
Centre-back Van de Ven converted into an offensive threat to be feared.
Tottenham's opening strike came from a cleverly executed set-piece play, with the midfielder directing the ball toward the six-yard box for an free the defender to head home.
And the captain - filling in as captain in the Argentine's unavailability - added his next goal in added time as he outjumped the England keeper to nod in.
Van de Ven was equally impressive in defence, partnering effectively with his defensive colleague as the pair dealt excellently with the 34 crosses thrown their way by Everton.
The club's new home had been a fortress, but it appears the freshness may have faded somewhat and fans headed to the gates after Sarr's third for Tottenham.
The end in Everton's unbeaten home run will not be a massive concern for the manager, but their attacking woes are.
Beto and the other striker - the Toffees' pair of recognised centre-forwards - had netted only one league goal combined before Sunday's outing.
And they were failed to contribute to that miserable tally on a wet and gloomy evening on in Liverpool.
The striker's performance was epitomized when he opted to bicycle-kick a clear chance from near the goal, despite having space and opportunity to readjust for a more conventional attempt that would have challenged the keeper.
Barry, as well, was unable to make an impact after replacing Beto in the 65th minute.
It was not for a lack of chance. The majority of the home side's offensive moves came through Grealish and the forward, who tirelessly delivered dangerous crosses into the Spurs penalty area.
The Toffees: The team head to the Stadium of Light in the top flight on Monday, 3 November at 20:00 GMT.
Tottenham: The London club go to St James' Park on Wednesday, 29 October for their fourth-round League Cup tie against Newcastle.
