Micky van de Ven became the initial defender to score twice in a English top-flight match for Tottenham since Jan Vertonghen in March 2013.
Match summary
Micky Van de Ven nods home twice in first half
The home side's defender has equalizer disallowed after video check
Pape Matar Sarr scores third in dying moments
This loss is Everton's initial at Hill Dickinson Stadium
Micky van de Ven found the net on two occasions as Tottenham prolonged their unbeaten run in away games in the top division and delivered the Toffees their initial loss at Hill Dickinson Stadium.
Everton were undefeated in their last five matches at their recently opened stadium at the dockside location, having departed Goodison Park behind after 133 years at the conclusion of the previous campaign.
But Spurs captain Van de Ven pounced 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 the midfielder - assisted by former Toffees attacker the Brazilian - nodded in a final goal to put the game to bed.
The Everton defender believed he had equalized the tally a short while 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, missed a number of chances, while the Italian goalkeeper made a series of fine saves to ensure the London club went five away games unbeaten in the English top flight.
But the video assistant referee intervened, advising the match official to review the goal on the pitchside monitor and the official decided that Jack Grealish and Iliman Ndiaye were in offside positions and had interfered with the Tottenham goalkeeper.
The win moves Thomas Frank's side up to third in the Premier League, a handful of points off north London rivals and leaders the Gunners.
Everton stay in 14th position and sit six points ahead of the relegation zone.
Following a drab midweek performance in the European competition against the French side, where they mustered a only attempts on goal, Frank opted for Randal Kolo Muani up front instead of ex-Everton forward Richarlison.
The summer signing was failed to make much of an impact before being replaced for Richarlison after an hour on the field, but it emerged he proved unnecessary in the end.
Defender Van de Ven transformed into an attacking force to be reckoned with.
Spurs' opening strike came from a cleverly executed corner routine, with Rodrigo Bentancur heading the ball back across the six-yard box for an unmarked Van de Ven to score.
And the captain - filling in as skipper in Cristian Romero's absence - notched his second in added time as he rose above Jordan Pickford to nod in.
The Dutchman was just as outstanding in defence, combining well with Kevin Danso as the pair dealt superbly with the 34 crosses sent their way by Everton.
Everton's new home had been a fortress, but it appears the freshness may have faded slightly and supporters flocked to the gates after the midfielder's third for Tottenham.
The break in Everton's winning streak at home won't be a major concern for the manager, but their attacking woes will be.
The forward and Thierno Barry - the Everton's pair of established centre-forwards - had scored only a single goal between them before the weekend's match.
And they were failed to add to that poor tally on a wet and gloomy night on Merseyside.
Beto's form was epitomized when he opted to overhead kick a clear chance from near the goal, despite having space and opportunity to readjust for a more conventional attempt that would have tested Vicario.
Barry, as well, failed to influence the game after coming on for the starter in the 65th minute.
It was not for a shortage of chance. Most of the home side's attacks came through the winger and Ndiaye, who tirelessly whipped threatening balls into the Tottenham box.
Everton: The team travel to Sunderland in the Premier League on Monday, 3 November at 20:00 GMT.
Spurs: The London club travel to St James' Park on late October for their next round League Cup match against the Magpies.
Liam Verhoeven is a digital content curator who scours the web for the most engaging reels and videos.