Micky van de Ven became the first backline player to score two goals in a English top-flight game for Tottenham since Jan Vertonghen in March 2013.
Key points
Micky Van de Ven nods into the net two times in first half
The home side's Jake O'Brien has equalizer disallowed after video check
Pape Matar Sarr scores final goal in 89th minute
The defeat is the Merseyside club's first at their new ground
The Tottenham defender found the net on two occasions as Spurs extended their unbeaten run on the road in the Premier League and delivered the Toffees their first defeat at their new home.
The Toffees were undefeated in their five previous fixtures at their new ground at the dockside location, having departed Goodison Park behind after more than a century at the conclusion of last season.
But Tottenham's leader Van de Ven struck twice from set-pieces in the opening 45 minutes to place their impressive record at risk, and they were unable to mount a comeback before the midfielder - set up by former Toffees attacker the Brazilian - nodded in a final goal to put the game to bed.
The Everton defender believed he had equalized the scores a short while after the defender's opener, directing a bullet header past Guglielmo Vicario from an set-piece.
Beto, in the meantime, failed to convert a handful of opportunities, while Vicario made a series of excellent stops to guarantee the London club went five away games unbeaten in the English top flight.
But the VAR stepped in, instructing Craig Pawson to check the score on the screen and the official ruled that Jack Grealish and the attacker were in offside positions and had impeded with the Tottenham goalkeeper.
The win moves the manager's team up to third in the division, five points off north London rivals and leaders the Gunners.
The Merseyside club remain in 14th position and are half a dozen points clear of the relegation zone.
Following a uninspiring weekday display in the European competition against Monaco, where they managed a only two shots on target, the manager chose the French striker in attack instead of former Toffees player Richarlison.
The summer signing was unable to influence the game before being replaced for the substitute after an hour on the pitch, but it emerged he wasn't needed in the end.
Centre-back Van de Ven transformed into an offensive threat to be feared.
Tottenham's opening strike came from a cleverly executed corner routine, with the midfielder heading the ball back across the face of goal for an unmarked the defender to score.
And the captain - filling in as skipper in Cristian Romero's absence - notched his second in first-half stoppage time as he rose above the England keeper to head home.
The Dutchman was just as outstanding in defence, combining well with Kevin Danso as the duo handled superbly with the numerous deliveries sent their way by the opposition.
Everton's recent stadium had been a happy hunting ground, but it appears the novelty may have faded slightly and fans headed to the exits after the midfielder's final goal for Spurs.
The break in Everton's unbeaten home run will not be a massive worry for the manager, but their attacking woes will be.
Beto and Thierno Barry - the Everton's pair of recognised strikers - had scored just one league goal between them before the weekend's match.
And they were failed to add to that miserable record on a wet and gloomy evening on in Liverpool.
The striker's form was summed up when he opted to overhead kick a clear chance from close range, despite having room and time to reposition for a simpler attempt that would have tested Vicario.
Barry, too, failed to make an impact after coming on for Beto in the 65th minute.
It wasn't for a lack of opportunity. Most of the home side's attacks came through the winger and Ndiaye, who tirelessly whipped dangerous balls into the Spurs box.
The Toffees: The team travel to Sunderland in the Premier League on Monday, 3 November at 8 PM.
Tottenham: The London club go to St James' Park on late October for their next round League Cup match against the Magpies.