Micky van de Ven became the first defender to net twice in a English top-flight game for Spurs since Jan Vertonghen in March 2013.
Match summary
The Spurs defender heads home two times in opening period
The home side's Jake O'Brien has equalizer disallowed after video check
Pape Matar Sarr scores third in dying moments
The defeat is Everton'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 Hill Dickinson Stadium.
Everton were unbeaten in their last five matches at their new ground at Bramley-Moore Dock, having left their historic home behind after more than a century at the conclusion of the previous campaign.
But Tottenham's leader Van de Ven pounced on two occasions from set-pieces in the opening 45 minutes to put their strong form at risk, and they were failed to turn it around before the midfielder - assisted by ex-Everton forward the Brazilian - nodded in a final goal to put the game to bed.
The Everton defender thought he had equalized the tally a short while after Van de Ven's opener, directing a powerful header past Guglielmo Vicario from an set-piece.
The Portuguese striker, in the meantime, missed a number of chances, while Vicario made a series of excellent stops to ensure Tottenham went five away games unbeaten in the Premier League.
But the VAR stepped in, instructing Craig Pawson to review the score on the screen and the referee ruled that Jack Grealish and Iliman Ndiaye were in offside positions and had impeded with the Tottenham goalkeeper.
This victory elevates the manager's side up to third in the Premier League, five points off local foes and leaders the Gunners.
The Merseyside club remain in 14th position and are six points clear of the drop zone.
Following a uninspiring midweek display in the European competition against the French side, where they mustered a mere two shots on target, Frank chose the French striker in attack instead of former Toffees player Richarlison.
The summer signing was failed to influence the game before being substituted for Richarlison after 60 minutes on the pitch, but it emerged he wasn't needed in the final outcome.
Defender Van de Ven converted into an attacking force to be reckoned with.
Tottenham's opening strike came from a cleverly executed corner routine, with the midfielder heading the ball back across the six-yard box for an free Van de Ven to head home.
And the captain - filling in as skipper in the Argentine's absence - added his next goal in first-half stoppage time as he rose above the England keeper to nod in.
The Dutchman was equally outstanding in defence, partnering effectively with his defensive colleague as the duo dealt excellently with the 34 crosses sent their way by the opposition.
The club's new home had been a fortress, but it appears the novelty may have worn off somewhat and fans headed to the exits after Sarr's third for Spurs.
The break in the team's winning streak at home won't be a massive concern for the manager, but their striker situation will be.
The forward and the other striker - the Everton's pair of established centre-forwards - had netted only a single goal combined before the weekend's outing.
And they were unable to contribute to that miserable tally on a damp and dreary night on Merseyside.
Beto's performance was summed up when he opted to overhead kick a golden opportunity from close range, despite having space and time to readjust for a more conventional attempt that would have challenged Vicario.
Barry, too, failed to influence the game after coming on for the starter in the second half.
It wasn't for a shortage of opportunity. Most of the home side's attacks came through Grealish and the forward, who tirelessly delivered threatening balls into the Tottenham penalty area.
The Toffees: The team head to the Stadium of Light in the Premier League on early November at 8 PM.
Tottenham: The London club go to Newcastle's ground on late October for their next round League Cup match against the Magpies.
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