Micky van de Ven emerged as the first backline player to score two goals in a Premier League game for Tottenham since Jan Vertonghen in March 2013.
Match summary
Micky Van de Ven nods into the net two times in opening period
The home side's Jake O'Brien has equalizer ruled after VAR review
Senegalese midfielder scores third in dying moments
The defeat is the Merseyside club's initial at their new ground
Micky van de Ven found the net on two occasions as Tottenham extended their unbeaten run on the road in the top division and handed the Toffees their first defeat at Hill Dickinson Stadium.
The Toffees were unbeaten in their last five matches at their recently opened stadium at Bramley-Moore Dock, having left Goodison Park behind after more than a century at the conclusion of the previous campaign.
But Spurs captain the Dutchman struck on two occasions from corners in the opening 45 minutes to put their impressive record at risk, and they were failed to turn it around before Pape Matar Sarr - set up by former Toffees attacker the Brazilian - headed home a final goal to put the game to bed.
Jake O'Brien believed he had equalized the tally a short while after Van de Ven's opener, directing a powerful header past the Spurs keeper from an Everton corner.
Beto, meanwhile, missed a handful of chances, while Vicario made a series of excellent saves to guarantee the London club went five matches on the road unbeaten in the English top flight.
But the video assistant referee stepped in, advising the match official to review the goal on the screen and the official ruled that the winger and Iliman Ndiaye were in offside positions and had interfered with the Tottenham goalkeeper.
The win elevates Thomas Frank's team up to third in the Premier League, five points off north London rivals and table-toppers Arsenal.
Everton remain in 14th position and are half a dozen points ahead of the drop zone.
Following a uninspiring weekday display in the Champions League against Monaco, where they managed a mere two shots on target, the manager opted for Randal Kolo Muani in attack instead of former Toffees player Richarlison.
The summer signing was failed to make much of an impact before being substituted for the substitute after an hour on the field, but it emerged he proved unnecessary in the end.
Defender the Dutchman converted into an offensive threat to be reckoned with.
Tottenham's opening strike came from a well-worked corner routine, with the midfielder heading the ball back across the face of goal for an unmarked Van de Ven to head home.
And the captain - stepping up as skipper in the Argentine's absence - added his second in added time as he outjumped the England keeper to nod in.
Van de Ven was equally impressive in defensive duties, combining effectively with Kevin Danso as the pair handled superbly with the 34 crosses thrown their way by the opposition.
The club's new home had been a happy hunting ground, but it seems the freshness may have faded slightly and fans flocked to the gates after Sarr's final goal for Tottenham.
The break in Everton's winning streak at home will not be a massive worry for the manager, but their attacking woes will be.
The forward and Thierno Barry - the Toffees' pair of established centre-forwards - had netted just one league goal combined before the weekend's outing.
And they were unable to add to that poor record on a damp and dreary night on in Liverpool.
The striker's form was summed up when he chose to overhead kick a golden opportunity from near the goal, even with having room and time 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 wasn't for a lack of opportunity. Most of the home side's attacks came through the winger and the forward, who tirelessly whipped threatening crosses into the Spurs penalty area.
The Toffees: Moyes' men head to the Stadium of Light in the Premier League on Monday, 3 November at 8 PM.
Spurs: The London club travel to Newcastle's ground on late October for their next round Carabao Cup match against Newcastle.
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