Micky van de Ven emerged as the initial backline player to score two goals in a Premier League match for Tottenham since Jan Vertonghen in March 2013.
Match summary
Micky Van de Ven heads home twice in first half
The home side's Jake O'Brien has goal disallowed after VAR review
Senegalese midfielder scores final goal in dying moments
The defeat is the Merseyside club's first at Hill Dickinson Stadium
Micky van de Ven scored twice as Tottenham prolonged their winning streak 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 recently opened stadium at the dockside location, having departed Goodison Park behind after 133 years at the end of the previous campaign.
But Tottenham's leader the Dutchman struck twice from set-pieces in the first half to place their strong form at jeopardy, and they were failed to mount a comeback before the midfielder - set up by former Toffees attacker the Brazilian - headed home a final goal to seal the victory.
The Everton defender thought he had levelled the scores a short while after Van de Ven's first goal, directing a bullet header past Guglielmo Vicario from an set-piece.
The Portuguese striker, meanwhile, missed a number of opportunities, while Vicario made a series of excellent stops to ensure Tottenham went five matches on the road unbeaten in the English top flight.
But the VAR intervened, advising Craig Pawson to review the goal on the screen and the referee ruled that Jack Grealish and the attacker were in offside positions and had impeded with the Spurs keeper.
The win elevates the manager's side up to third in the Premier League, five points off local foes and table-toppers Arsenal.
Everton stay 14th and are half a dozen points clear of the relegation zone.
Following a drab midweek performance in the European competition against the French side, where they managed a mere two shots on goal, Frank opted for Randal Kolo Muani up front instead of ex-Everton forward the Brazilian.
The summer signing was failed to make much of an impact before being substituted for the substitute after an hour on the pitch, but it turns out he proved unnecessary in the end.
Defender the Dutchman converted into an offensive threat to be feared.
Tottenham's first goal came from a well-worked corner routine, with the midfielder heading the ball toward the face of goal for an free the defender to head home.
And the captain - 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 Kevin Danso as the duo dealt excellently with the 34 crosses thrown their way by Everton.
Everton's recent stadium had been a fortress, but it appears the freshness may have worn off somewhat and fans headed to the gates after the midfielder's third for Tottenham.
The break in the team's winning streak at home won't be a massive worry for David Moyes, but their attacking woes are.
The forward and the other striker - the Toffees' pair of established centre-forwards - had netted just a single goal between them before the weekend's outing.
And they were unable to contribute to that poor record on a damp and dreary night on in Liverpool.
The striker's form was summed up when he chose to bicycle-kick a clear chance from close range, even with having room and time to readjust for a simpler shot that would have challenged the keeper.
Barry, as well, was unable to influence the game after coming on for Beto in the 65th minute.
It wasn't for a lack of opportunity. Most of Everton's offensive moves came through the winger and the forward, who tirelessly whipped dangerous balls into the Spurs penalty area.
Everton: The team travel to Sunderland in the top flight on early November at 20:00 GMT.
Spurs: Spurs travel to Newcastle's ground on late October for their next round League Cup match against Newcastle.
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