Micky van de Ven emerged as the initial defender to net two goals in a Premier League game for Tottenham since Jan Vertonghen in March 2013.
At a glance
Micky Van de Ven heads home two times in first half
Everton's defender has equalizer disallowed after video check
Pape Matar Sarr nets third in 89th minute
This loss is the Merseyside club's initial at Hill Dickinson Stadium
The Tottenham defender scored twice as Tottenham prolonged their unbeaten run on the road in the top division and handed Everton their initial loss at Hill Dickinson Stadium.
The Toffees were unbeaten in their five previous fixtures at their recently opened stadium at the dockside location, having departed their historic home behind after 133 years at the conclusion of the previous campaign.
But Tottenham's leader the Dutchman struck on two occasions from set-pieces in the first half to put their strong form at jeopardy, and they were failed to turn it around before the midfielder - assisted by former Toffees attacker Richarlison - headed home a final goal to put the game to bed.
Jake O'Brien believed he had levelled the tally five minutes after Van de Ven's opener, glancing a bullet header past the Spurs keeper from an Everton corner.
The Portuguese striker, meanwhile, missed a handful of opportunities, while Vicario made a succession 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 intervened, advising Craig Pawson to check the score on the pitchside monitor and the referee decided that the winger and Iliman Ndiaye were in an illegal position and had impeded with the Tottenham goalkeeper.
This victory moves the manager's side up to third in the Premier League, five points off north London rivals and leaders Arsenal.
The Merseyside club stay 14th and sit half a dozen points clear of the drop zone.
Following a uninspiring midweek display in the Champions League against Monaco, where they managed a only two shots on goal, the manager chose Randal Kolo Muani up front instead of ex-Everton forward Richarlison.
The new arrival was failed to influence the game before being substituted for the substitute 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.
Tottenham's first goal came from a cleverly executed set-piece play, with Rodrigo Bentancur directing the ball back across the six-yard box for an unmarked Van de Ven to head home.
And Ven de Ven - stepping up as skipper in Cristian Romero's absence - notched his next goal in first-half stoppage time as he outjumped the England keeper to head home.
Van de Ven was just as impressive in defensive duties, partnering effectively with Kevin Danso as the duo dealt excellently with the 34 crosses sent their way by Everton.
Everton's new home had been a fortress, but it seems the novelty may have worn off somewhat and supporters headed to the exits after the midfielder's final goal for Tottenham.
The break in Everton's winning streak at home will not be a major concern for the manager, but their attacking woes are.
The forward and the other striker - the Toffees' pair of established strikers - had netted just a single goal combined before Sunday's outing.
And they were unable to add to that poor tally on a wet and gloomy night on in Liverpool.
Beto's performance was epitomized when he opted to bicycle-kick a golden opportunity from close range, despite having room and opportunity to readjust for a more conventional attempt that would have challenged Vicario.
Barry, as well, was unable to make an impact after coming on for Beto in the second half.
It wasn't for a shortage of chance. The majority of the home side's offensive moves came through Grealish and Ndiaye, who tirelessly whipped dangerous crosses into the Tottenham box.
The Toffees: Moyes' men head to Sunderland in the top flight on Monday, 3 November at 8 PM.
Spurs: Spurs travel to Newcastle's ground on Wednesday, 29 October for their next round League Cup match against Newcastle.