Micky van de Ven became the initial backline player to score two goals in a English top-flight game for Spurs since the Belgian defender in spring 2013.
At a glance
The Spurs defender heads home two times in first half
The home side's defender has equalizer ruled after VAR review
Pape Matar Sarr scores third in dying moments
This loss is the Merseyside club's first at Hill Dickinson Stadium
Micky van de Ven found the net on two occasions as Spurs extended their winning streak on the road in the Premier League and handed the Toffees their initial loss at Hill Dickinson Stadium.
The Toffees were undefeated in their five previous fixtures at their new ground at Bramley-Moore Dock, having left their historic home behind after more than a century at the end of last season.
But Tottenham's leader the Dutchman pounced twice from set-pieces in the opening 45 minutes to put their impressive record at risk, and they were failed to turn it around before Pape Matar Sarr - assisted by ex-Everton forward the Brazilian - nodded in a final goal to put the game to bed.
The Everton defender thought he had levelled the scores a short while after the defender's opener, glancing a powerful header past the Spurs keeper from an Everton corner.
Beto, in the meantime, failed to convert a handful of chances, while Vicario made a series of fine saves to ensure Tottenham went five matches on the road unbeaten in the Premier League.
But the VAR stepped in, advising the match official to check the goal on the screen and the referee decided that the winger and Iliman Ndiaye were in an illegal position and had impeded with the Tottenham goalkeeper.
The win elevates Thomas Frank's side into third place in the division, a handful of points off north London rivals and leaders the Gunners.
Everton stay in 14th position and are half a dozen points clear of the relegation zone.
Following a uninspiring midweek display in the European competition against the French side, where they managed a only attempts on target, Frank chose the French striker up front instead of ex-Everton forward Richarlison.
The new arrival was unable to make much of an impact before being substituted for the substitute after 60 minutes on the field, but it emerged he wasn't needed in the end.
Centre-back Van de Ven converted into an offensive threat to be feared.
Tottenham's opening strike came from a well-worked corner routine, with the midfielder directing the ball back across the face of goal for an free the defender to score.
And Ven de Ven - stepping up as skipper in the Argentine's absence - added his second in added time as he outjumped the England keeper to nod in.
Van de Ven was just as impressive in defensive duties, partnering effectively with his defensive colleague as the duo handled superbly with the 34 crosses sent their way by the opposition.
Everton's recent stadium had been a fortress, but it appears the novelty may have faded somewhat and supporters flocked to the exits after Sarr's third for Tottenham.
The end in Everton's unbeaten home run will not be a massive worry for David Moyes, but their attacking woes are.
Beto and Thierno Barry - the Everton's pair of recognised centre-forwards - had netted only a single goal between them before Sunday's match.
And they were unable to contribute to that miserable tally on a wet and gloomy evening on in Liverpool.
Beto's form was epitomized when he chose to bicycle-kick a clear chance from close range, despite having space and time to reposition for a more conventional shot that would have tested the keeper.
Barry, too, failed to make an impact after replacing Beto in the second half.
It wasn't for a lack of opportunity. The majority of the home side's offensive moves came through the winger and Ndiaye, who tirelessly delivered dangerous balls into the Spurs box.
Everton: The team head to Sunderland in the Premier League on Monday, 3 November at 8 PM.
Tottenham: The London club go to St James' Park on Wednesday, 29 October for their next round League Cup match against Newcastle.
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