The Dutch defender became the initial backline player to score two goals in a Premier League game for Tottenham since the Belgian defender in March 2013.
Match summary
The Spurs defender heads into the net twice in opening period
Everton's defender has equalizer disallowed after VAR review
Pape Matar Sarr nets third in dying moments
This loss is Everton's first at their new ground
Micky van de Ven scored twice as Spurs extended their unbeaten run in away games in the top division and delivered Everton their initial loss 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 133 years at the end of last season.
But Spurs captain Van de Ven struck twice from corners in the opening 45 minutes to place their impressive record at jeopardy, and they were unable to turn it around before the midfielder - assisted by former Toffees attacker the Brazilian - nodded in a third to put the game to bed.
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, meanwhile, failed to convert a handful of opportunities, while Vicario made a succession of excellent saves to ensure the London club went five away games without defeat in the Premier League.
But the VAR 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 offside positions and had impeded with the Spurs keeper.
The win elevates the manager's team into third place in the Premier League, a handful of points off north London rivals and leaders the Gunners.
Everton remain 14th and are six points clear of the relegation zone.
Following a drab midweek display in the European competition against Monaco, where they managed a mere two shots on goal, Frank opted for Randal Kolo Muani in attack instead of ex-Everton forward the Brazilian.
The summer signing was unable to influence the game before being replaced for Richarlison 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 the defender to head home.
And the captain - filling in as skipper in the Argentine's absence - notched his second in added time as he rose above Jordan Pickford to nod in.
The Dutchman was equally outstanding in defensive duties, combining well with his defensive colleague as the pair handled superbly with the 34 crosses sent their way by Everton.
The club's recent stadium had been a happy hunting ground, but it appears the freshness may have faded slightly and fans headed to the gates after the midfielder's final goal for Tottenham.
The break in the team's winning streak at home won't be a major worry for David Moyes, but their striker situation will be.
Beto and Thierno Barry - the Everton's pair of recognised strikers - had scored just one league goal between them before Sunday's match.
And they were failed to add to that miserable tally on a wet and gloomy night on in Liverpool.
The striker's performance was epitomized when he opted to bicycle-kick a clear chance from close range, even with having room and opportunity to reposition for a more conventional shot that would have tested Vicario.
The substitute, as well, was unable to make an impact after replacing Beto in the second half.
It wasn't for a shortage of chance. Most of Everton's offensive moves came through Grealish and the forward, who tirelessly delivered threatening crosses into the Tottenham penalty area.
Everton: Moyes' men travel to Sunderland in the Premier League on Monday, 3 November at 8 PM.
Spurs: Spurs travel to Newcastle's ground on late October for their fourth-round Carabao Cup match against the Magpies.
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