Micky van de Ven emerged as the first backline player to net twice in a Premier League match for Tottenham since Jan Vertonghen in spring 2013.
Match summary
The Spurs defender nods 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
This loss is Everton's first at Hill Dickinson Stadium
Micky van de Ven found the net on two occasions as Spurs prolonged their unbeaten run in away games in the top division and delivered the Toffees their initial loss at their new home.
The Toffees were unbeaten in their five previous fixtures at their new ground at Bramley-Moore Dock, having left their historic home behind after more than a century at the end of the previous campaign.
But Tottenham's leader the Dutchman pounced on two occasions from corners in the first half to place their impressive record at jeopardy, and they were failed to turn it around before the midfielder - assisted by ex-Everton forward the Brazilian - nodded in a third to put the game to bed.
The Everton defender thought he had levelled the scores five minutes after Van de Ven's first goal, directing a bullet header past Guglielmo Vicario from an set-piece.
Beto, in the meantime, failed to convert a number of chances, while Vicario made a series of fine saves to ensure the London club went five matches on the road unbeaten in the English top flight.
But the VAR intervened, advising the match official to review the score on the pitchside monitor and the referee decided that Jack Grealish and Iliman Ndiaye were in offside positions and had impeded with the Spurs keeper.
This victory moves the manager's team into third place in the Premier League, five points off north London rivals and table-toppers the Gunners.
The Merseyside club remain in 14th position and are six points clear of the relegation zone.
After a uninspiring weekday display in the Champions League against Monaco, where they mustered a mere two shots on target, the manager chose the French striker in attack instead of former Toffees player the Brazilian.
The new arrival was failed to make much of an impact before being replaced for the substitute after an hour on the field, but it emerged he wasn't needed in the final outcome.
Defender Van de Ven converted into an attacking force to be feared.
Tottenham's opening strike came from a well-worked set-piece play, with Rodrigo Bentancur directing the ball toward the face of goal for an unmarked the defender to head home.
And the captain - stepping up as skipper in the Argentine's unavailability - notched his second in added time as he outjumped Jordan Pickford to head home.
The Dutchman was equally impressive in defensive duties, combining well with Kevin Danso as the pair dealt superbly with the 34 crosses sent their way by the opposition.
Everton's new home had been a happy hunting ground, but it seems the freshness may have worn off slightly and fans flocked to the gates after Sarr's third for Spurs.
The end in Everton's unbeaten home run will not be a major concern for the manager, but their striker situation will be.
The forward and the other striker - the Toffees' pair of established strikers - had scored only one league goal between them before Sunday's outing.
And they were failed to contribute to that miserable tally on a damp and dreary night on in Liverpool.
The striker's form was epitomized when he opted to bicycle-kick a clear chance from close range, even with having space and opportunity to readjust for a more conventional attempt that would have challenged the keeper.
The substitute, as well, was unable to influence the game after replacing Beto in the second half.
It was not for a shortage of chance. The majority of the home side's attacks came through the winger and Ndiaye, who relentlessly delivered threatening balls into the Spurs penalty area.
The Toffees: Moyes' men travel to Sunderland in the top flight on early November at 20:00 GMT.
Spurs: Spurs travel to St James' Park on Wednesday, 29 October for their next round League Cup tie against Newcastle.
Elena Voss is a tech enthusiast and writer with over a decade of experience in software development and digital media.