Micky van de Ven became the first backline player to net twice in a English top-flight match for Tottenham since the Belgian defender in spring 2013.
Match summary
The Spurs defender nods into the net twice in first half
The home side's defender has equalizer ruled after VAR review
Pape Matar Sarr scores third in dying moments
The defeat is the Merseyside club's initial at their new ground
The Tottenham defender scored on two occasions as Spurs prolonged their winning streak in away games in the top division and delivered Everton their initial loss at Hill Dickinson Stadium.
The Toffees were unbeaten in their last five matches at their new ground at the dockside location, having departed Goodison Park behind after more than a century at the end of the previous campaign.
But Spurs captain the Dutchman pounced on two occasions from set-pieces in the first half to put their impressive record 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 seal the victory.
The Everton defender believed he had levelled the scores a short while after the defender's first goal, directing a powerful header past Guglielmo Vicario from an set-piece.
Beto, meanwhile, failed to convert a handful of chances, while the Italian goalkeeper made a series of fine stops to guarantee Tottenham went five matches on the road without defeat in the Premier League.
But the VAR stepped in, instructing the match official to review the score on the pitchside monitor and the referee ruled that Jack Grealish and Iliman Ndiaye were in an illegal position and had interfered with the Spurs keeper.
This victory elevates the manager's team into third place in the Premier League, five points off local foes and table-toppers the Gunners.
The Merseyside club stay in 14th position and are half a dozen points clear of the relegation zone.
Following a uninspiring weekday performance in the Champions League against Monaco, where they managed a mere attempts on goal, Frank chose the French striker in attack instead of former Toffees player Richarlison.
The new arrival was unable to make much of an impact before being replaced for the substitute after 60 minutes on the field, but it emerged he wasn't needed in the final outcome.
Centre-back Van de Ven converted into an offensive threat to be reckoned with.
Spurs' first goal came from a cleverly executed set-piece play, with Rodrigo Bentancur heading the ball toward the face of goal for an unmarked Van de Ven to head home.
And Ven de Ven - filling in as skipper in the Argentine's unavailability - notched his next goal in added time as he rose above Jordan Pickford to head home.
The Dutchman was equally impressive in defence, combining well with Kevin Danso as the duo handled superbly with the numerous deliveries thrown their way by Everton.
The club's new home had been a fortress, but it seems the novelty may have faded slightly and supporters flocked to the exits after Sarr's final goal for Spurs.
The break in the team's winning streak at home won't be a massive concern for David Moyes, but their attacking woes are.
The forward and the other striker - the Everton's two established strikers - had netted only one league goal combined before the weekend's outing.
And they were failed to contribute to that poor tally on a wet and gloomy evening on Merseyside.
Beto's form was summed up when he opted to bicycle-kick a golden opportunity from near the goal, even with having room and time to readjust for a simpler attempt that would have tested Vicario.
Barry, as well, failed to make an impact after coming on for Beto in the second half.
It was not for a shortage of chance. The majority of Everton's attacks came through Grealish and the forward, who relentlessly delivered dangerous crosses into the Tottenham box.
Everton: Moyes' men head to the Stadium of Light in the top flight on Monday, 3 November at 20:00 GMT.
Spurs: The London club travel to St James' Park on Wednesday, 29 October for their next round League Cup tie against the Magpies.
A seasoned gambling analyst with over a decade of experience in casino operations and game strategy development.