Micky van de Ven emerged as the first backline player to score twice in a Premier League game for Spurs since Jan Vertonghen in spring 2013.
At a glance
The Spurs defender nods into the net two times in opening period
Everton's defender has equalizer ruled after VAR review
Senegalese midfielder scores third in 89th minute
The defeat is the Merseyside club's first at Hill Dickinson Stadium
Micky van de Ven scored on two occasions as Spurs prolonged their unbeaten run on the road in the top division and delivered the Toffees their first defeat at Hill Dickinson Stadium.
Everton were unbeaten in their last five matches at their new ground at the dockside location, having left their historic home behind after more than a century at the conclusion of last season.
But Tottenham's leader the Dutchman struck twice from set-pieces in the opening 45 minutes to place their strong form at risk, and they were failed to mount a comeback before Pape Matar Sarr - assisted by former Toffees attacker the Brazilian - nodded in a final goal to seal the victory.
Jake O'Brien believed he had levelled the scores a short while after Van de Ven's opener, directing a powerful header past the Spurs keeper from an Everton corner.
Beto, meanwhile, failed to convert a handful of chances, while the Italian goalkeeper made a series of excellent saves to guarantee Tottenham went five matches on the road unbeaten in the Premier League.
But the video assistant referee stepped in, advising the match official to check the goal on the pitchside monitor and the referee ruled that the winger and the attacker were in offside positions and had impeded with the Spurs keeper.
This victory elevates Thomas Frank's side into third place in the division, five points off north London rivals and leaders Arsenal.
The Merseyside club stay 14th and sit half a dozen points clear of the relegation zone.
After a uninspiring weekday performance in the European competition against Monaco, where they managed a mere two shots on target, the manager chose the French striker in attack instead of former Toffees player Richarlison.
The new arrival was failed to make much of an impact before being substituted for Richarlison after an hour on the field, but it turns out he wasn't needed in the end.
Defender the Dutchman transformed into an offensive threat to be reckoned with.
Tottenham's opening strike came from a cleverly executed corner routine, with Rodrigo Bentancur directing the ball back across the face of goal for an free Van de Ven to head home.
And the captain - filling in as skipper in the Argentine's unavailability - notched his second in first-half stoppage time as he outjumped the England keeper to head home.
The Dutchman was just as outstanding in defensive duties, partnering effectively with Kevin Danso as the pair handled excellently with the numerous deliveries sent their way by the opposition.
Everton's new home had been a happy hunting ground, but it appears the freshness may have worn off slightly and fans flocked to the gates after the midfielder's third for Spurs.
The break in Everton's unbeaten home run won't be a massive concern for David Moyes, but their attacking woes will be.
Beto and the other striker - the Toffees' pair of established strikers - had scored just a single goal combined before the weekend's match.
And they were unable to contribute to that poor record on a damp and dreary evening on Merseyside.
The striker's form was epitomized when he opted to bicycle-kick a clear chance from close range, despite having space and opportunity to reposition for a simpler attempt that would have tested the keeper.
Barry, too, failed to influence the game after coming on for Beto in the second half.
It was not for a shortage of opportunity. Most of the home side's attacks came through the winger and the forward, who relentlessly whipped threatening crosses into the Tottenham penalty area.
The Toffees: Moyes' men head to the Stadium of Light in the Premier League on early November at 8 PM.
Tottenham: Spurs travel to Newcastle's ground on Wednesday, 29 October for their next round League Cup tie against the Magpies.
Elena is a tech enthusiast and freelance writer specializing in gadget reviews and digital trends.