The Dutch defender emerged as the first backline player to score twice in a English top-flight game for Spurs since Jan Vertonghen in March 2013.
Match summary
Micky Van de Ven nods home two times in first half
The home side's defender has equalizer ruled after video check
Pape Matar Sarr nets final goal in 89th minute
The defeat is the Merseyside club's initial at their new ground
The Tottenham defender found the net twice as Spurs prolonged their unbeaten run in away games in the top division and handed Everton their initial loss at Hill Dickinson Stadium.
The Toffees were undefeated in their last five matches at their new ground at Bramley-Moore Dock, having left their historic home behind after 133 years at the end of the previous campaign.
But Tottenham's leader the Dutchman pounced twice from corners in the opening 45 minutes to place their impressive record at jeopardy, and they were failed to turn it around before Pape Matar Sarr - set up by former Toffees attacker Richarlison - nodded in a final goal to seal the victory.
The Everton defender believed he had equalized the scores a short while after the defender's first goal, directing a bullet header past the Spurs keeper from an set-piece.
The Portuguese striker, in the meantime, missed a number of opportunities, while Vicario made a series of fine stops to guarantee the London club went five away games unbeaten in the English top flight.
But the video assistant referee stepped in, advising the match official to review the score on the screen and the referee decided that Jack Grealish and the attacker were in an illegal position and had impeded with the Spurs keeper.
The win elevates the manager's team up to third in the division, a handful of points off north London rivals and table-toppers Arsenal.
The Merseyside club remain in 14th position and are six points clear of the relegation zone.
After a uninspiring midweek display in the European competition against the French side, where they managed a only attempts on target, Frank opted for Randal Kolo Muani up front instead of ex-Everton forward the Brazilian.
The summer signing was unable to influence the game before being substituted for the substitute after 60 minutes on the field, but it turns out he wasn't needed in the final outcome.
Defender the Dutchman transformed into an offensive threat to be reckoned with.
Spurs' first goal came from a cleverly executed corner routine, with the midfielder heading the ball toward the six-yard box for an free Van de Ven to head home.
And the captain - filling in as skipper in the Argentine's unavailability - notched his second in first-half stoppage time as he outjumped the England keeper to nod in.
Van de Ven was equally outstanding in defence, combining well with his defensive colleague as the duo handled excellently with the 34 crosses sent their way by the opposition.
The club's new home had been a fortress, but it seems the freshness may have faded somewhat and supporters flocked to the exits after the midfielder's final goal for Tottenham.
The end in Everton's winning streak at home will not be a massive worry for the manager, but their striker situation will be.
The forward and Thierno Barry - the Toffees' two established centre-forwards - had netted only a single goal between them before the weekend's match.
And they were failed to contribute to that poor tally on a damp and dreary evening on Merseyside.
Beto's form was summed up when he chose to overhead kick a golden opportunity from near the goal, even with having room and time to readjust for a simpler attempt that would have challenged the keeper.
The substitute, too, failed to make an impact after coming on for the starter in the second half.
It was not for a lack of opportunity. Most of Everton's attacks came through Grealish and the forward, who tirelessly delivered dangerous crosses into the Tottenham penalty area.
The Toffees: The team travel to the Stadium of Light in the Premier League on early November at 20:00 GMT.
Tottenham: The London club go to St James' Park on late October for their next round League Cup tie against the Magpies.
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