Micky van de Ven emerged as the initial backline player to net two goals in a English top-flight game for Tottenham since the Belgian defender in spring 2013.
Match summary
The Spurs defender heads home two times in first half
The home side's defender has goal disallowed after VAR review
Pape Matar Sarr scores final goal in dying moments
The defeat is the Merseyside club's initial at their new ground
The Tottenham defender scored on two occasions as Spurs extended their winning streak on the road in the top division and delivered the Toffees their first defeat at their new home.
Everton were unbeaten in their last five matches at their recently opened stadium at the dockside location, having departed Goodison Park behind after 133 years at the end of last season.
But Spurs captain the Dutchman struck on two occasions from set-pieces in the opening 45 minutes to put their impressive record at risk, and they were unable to turn it around before Pape Matar Sarr - assisted by former Toffees attacker the Brazilian - headed home a final goal to seal the victory.
Jake O'Brien believed he had equalized the scores a short while after the defender's opener, directing a powerful header past the Spurs keeper from an set-piece.
The Portuguese striker, meanwhile, missed a number of opportunities, while the Italian goalkeeper made a succession of excellent stops to ensure the London club went five matches on the road without defeat in the English top flight.
But the VAR intervened, instructing Craig Pawson to check the goal on the pitchside monitor and the referee ruled that the winger and Iliman Ndiaye were in an illegal position and had impeded with the Tottenham goalkeeper.
This victory moves Thomas Frank's team into third place in the division, five points off north London rivals and leaders Arsenal.
Everton stay in 14th position and are half a dozen points clear of the relegation zone.
Following a drab weekday performance in the Champions League against Monaco, where they managed a only attempts on target, the manager opted for Randal Kolo Muani in attack instead of former Toffees player the Brazilian.
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 turns out he proved unnecessary in the end.
Centre-back the Dutchman converted into an offensive threat to be reckoned with.
Spurs' first goal came from a well-worked corner routine, with Rodrigo Bentancur heading the ball back across the six-yard box for an free the defender to head home.
And the captain - stepping up as captain in the Argentine's absence - notched his second in first-half stoppage time as he rose above Jordan Pickford to head home.
Van de Ven was just as impressive in defence, combining well with his defensive colleague as the duo dealt excellently with the numerous deliveries sent their way by the opposition.
The club's new home had been a fortress, but it seems the novelty may have worn off somewhat and fans headed to the exits after the midfielder's third for Spurs.
The break in Everton's winning streak at home won't be a massive concern for the manager, but their striker situation are.
The forward and the other striker - the Everton's two established centre-forwards - had scored just one league goal combined before Sunday's outing.
And they were unable to contribute to that poor tally on a damp and dreary evening on in Liverpool.
Beto's form was summed up when he chose to bicycle-kick a clear chance from close range, despite having room and opportunity to reposition for a simpler shot that would have tested Vicario.
Barry, too, was unable to influence the game after coming on for the starter in the second half.
It was not for a lack of opportunity. The majority of the home side's attacks came through Grealish and Ndiaye, who relentlessly whipped dangerous crosses into the Tottenham penalty area.
The Toffees: Moyes' men head to the Stadium of Light in the Premier League on Monday, 3 November at 8 PM.
Tottenham: The London club travel to St James' Park on Wednesday, 29 October for their fourth-round Carabao Cup match against the Magpies.