Micky van de Ven emerged as the initial backline player to score twice in a English top-flight match for Tottenham since Jan Vertonghen in spring 2013.
At a glance
Micky Van de Ven nods home two times in first half
Everton's defender has equalizer disallowed after VAR review
Pape Matar Sarr nets third in dying moments
The defeat is the Merseyside club's initial at their new ground
The Tottenham defender found the net on two occasions as Tottenham prolonged their unbeaten run on the road in the top division and delivered Everton their first defeat at their new home.
Everton were unbeaten in their last five matches at their new ground at Bramley-Moore Dock, having departed Goodison Park behind after more than a century at the end of the previous campaign.
But Tottenham's leader Van de Ven struck twice from set-pieces in the first half to put their strong form at jeopardy, and they were failed to turn it around before Pape Matar Sarr - set up by former Toffees attacker the Brazilian - headed home a third to seal the victory.
Jake O'Brien believed he had equalized the scores a short while after the defender's first goal, directing a powerful header past Guglielmo Vicario from an Everton corner.
Beto, in the meantime, missed a number of opportunities, while the Italian goalkeeper made a succession of excellent saves to ensure Tottenham went five away games unbeaten in the Premier League.
But the VAR stepped in, advising Craig Pawson to review the goal on the screen and the referee decided that Jack Grealish and the attacker were in offside positions and had impeded with the Tottenham goalkeeper.
The win moves the manager's side up to third in the Premier League, a handful of points off north London rivals and leaders the Gunners.
The Merseyside club remain in 14th position and are six points clear of the drop zone.
After a drab weekday display in the European competition against the French side, where they mustered a only attempts on target, the manager opted for the French striker up front instead of former Toffees player the Brazilian.
The summer signing was unable to influence the game before being substituted for the substitute after an hour on the field, but it emerged he wasn't needed in the final outcome.
Defender Van de Ven transformed into an attacking force to be reckoned with.
Spurs' first goal came from a cleverly executed set-piece play, with the midfielder directing the ball back across the face of goal for an unmarked Van de Ven to score.
And the captain - filling in as skipper in the Argentine's absence - notched his second in added time as he outjumped the England keeper to head home.
Van de Ven was just as impressive in defensive duties, combining effectively with Kevin Danso as the duo dealt excellently with the numerous deliveries thrown their way by the opposition.
The club's recent stadium had been a fortress, but it appears the freshness may have faded somewhat and supporters headed to the exits after Sarr's final goal for Tottenham.
The break in the team's winning streak at home will not be a massive concern for David Moyes, but their attacking woes will be.
The forward and the other striker - the Everton's two recognised centre-forwards - had netted only a single goal between them before the weekend's match.
And they were unable to contribute to that poor record on a wet and gloomy night on in Liverpool.
Beto's performance was summed up when he chose to bicycle-kick a clear chance from close range, despite having space and time to readjust for a simpler attempt that would have tested Vicario.
Barry, as well, failed to influence the game after replacing the starter in the second half.
It was not for a shortage of chance. Most of Everton's offensive moves came through Grealish and the forward, who relentlessly delivered threatening crosses into the Spurs penalty area.
Everton: Moyes' men head to Sunderland in the Premier League on Monday, 3 November at 20:00 GMT.
Tottenham: The London club go to St James' Park on Wednesday, 29 October for their next round Carabao Cup match against Newcastle.
Eleanor is a passionate writer and cultural enthusiast, sharing her experiences and discoveries from across the UK.