Micky van de Ven emerged as the first backline player to net two goals in a Premier League game for Spurs since the Belgian defender in spring 2013.
At a glance
The Spurs defender heads home twice in first half
Everton's Jake O'Brien has equalizer ruled after video check
Pape Matar Sarr scores final goal in dying moments
The defeat is the Merseyside club's first at Hill Dickinson Stadium
Micky van de Ven scored twice as Tottenham extended their unbeaten run on the road in the top division and handed Everton their first defeat at Hill Dickinson Stadium.
The Toffees were unbeaten in their five previous fixtures at their new ground at the dockside location, having departed their historic home behind after 133 years at the conclusion of last season.
But Spurs captain the Dutchman pounced on two occasions from corners in the first half to put their impressive record at risk, and they were unable to turn it around before the midfielder - set up by former Toffees attacker Richarlison - nodded in a final goal to put the game to bed.
The Everton defender 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.
Beto, in the meantime, missed a handful of chances, while Vicario made a succession of excellent saves to guarantee Tottenham went five matches on the road without defeat in the Premier League.
But the video assistant referee intervened, instructing the match official to review the goal on the screen and the official ruled that the winger and the attacker were in offside positions and had impeded with the Spurs keeper.
The win elevates the manager's team into third place in the Premier League, five points off north London rivals and table-toppers Arsenal.
Everton stay in 14th position and are six points ahead of the drop zone.
After a drab midweek display in the European competition against the French side, where they managed a mere two shots on goal, Frank opted for Randal Kolo Muani up front instead of ex-Everton forward the Brazilian.
The new arrival was unable to make much of an impact before being substituted for Richarlison after an hour on the field, but it turns out he wasn't needed in the final outcome.
Defender Van de Ven transformed into an offensive threat to be feared.
Spurs' first goal came from a cleverly executed set-piece play, with Rodrigo Bentancur directing the ball toward the face of goal for an free the defender to score.
And the captain - filling in as captain in Cristian Romero's absence - notched his next goal in added time as he outjumped the England keeper to nod in.
The Dutchman was equally outstanding in defence, partnering well with his defensive colleague as the duo handled excellently with the 34 crosses sent their way by the opposition.
The club's new home had been a fortress, but it seems the novelty may have faded slightly and supporters flocked to the exits after Sarr's third for Tottenham.
The end in the team's winning streak at home will not be a massive worry for David Moyes, but their attacking woes are.
Beto and Thierno Barry - the Toffees' two established centre-forwards - had scored only one league goal combined before Sunday's match.
And they were failed to add to that poor record on a damp and dreary night on in Liverpool.
Beto's form was epitomized when he opted to overhead kick a clear chance from near the goal, even with having space and opportunity to readjust for a simpler shot that would have challenged the keeper.
The substitute, too, was unable to make an impact after coming on for Beto in the 65th minute.
It wasn't for a lack of chance. The majority of Everton's offensive moves came through Grealish and the forward, who tirelessly delivered threatening balls into the Spurs penalty area.
The Toffees: Moyes' men travel to Sunderland in the top flight on Monday, 3 November at 20:00 GMT.
Spurs: The London club go to Newcastle's ground on late October for their next round Carabao Cup match against the Magpies.
Mira is a tech journalist and AI researcher with over a decade of experience covering emerging technologies and their societal impacts.