Micky van de Ven became the first defender to net two goals in a English top-flight match for Spurs since Jan Vertonghen in spring 2013.
Key points
Micky Van de Ven heads into the net twice in first half
The home side's Jake O'Brien has goal disallowed after video check
Pape Matar Sarr nets final goal in 89th minute
The defeat is Everton's first at their new ground
Micky van de Ven found the net on two occasions as Tottenham prolonged their unbeaten run on the road in the top division and handed Everton their first defeat at Hill Dickinson Stadium.
The Toffees were undefeated in their last five matches at their new ground at the dockside location, having departed Goodison Park behind after more than a century at the end of last season.
But Spurs captain the Dutchman struck twice from corners in the opening 45 minutes to put their impressive record at jeopardy, and they were failed to turn it around before the midfielder - set up by former Toffees attacker Richarlison - headed home a third to put the game to bed.
Jake O'Brien believed he had equalized the tally a short while after the defender's first goal, directing a powerful header past Guglielmo Vicario from an Everton corner.
The Portuguese striker, in the meantime, failed to convert a number of chances, while Vicario made a succession of excellent stops to ensure Tottenham went five matches on the road without defeat in the English top flight.
But the video assistant referee stepped in, instructing Craig Pawson to review the goal on the pitchside monitor and the referee decided that the winger and Iliman Ndiaye were in an illegal position and had interfered with the Spurs keeper.
This victory moves the manager's team into third place in the Premier League, five points off local foes and table-toppers Arsenal.
Everton remain in 14th position and are half a dozen points clear of the drop zone.
After a uninspiring midweek performance in the Champions League against the French side, where they mustered a only two shots on goal, Frank chose the French striker up front instead of ex-Everton forward Richarlison.
The summer signing was failed to make much of an impact before being substituted for Richarlison after an hour on the pitch, but it emerged he wasn't needed in the end.
Centre-back the Dutchman transformed into an attacking force to be reckoned with.
Spurs' first goal came from a cleverly executed corner routine, with the midfielder heading the ball back across the face of goal for an unmarked Van de Ven to score.
And Ven de Ven - filling in as captain in Cristian Romero's absence - notched his second in added time as he rose above Jordan Pickford to head home.
Van de Ven was equally outstanding in defence, combining well with his defensive colleague as the pair dealt excellently with the numerous deliveries thrown their way by Everton.
Everton's recent stadium had been a happy hunting ground, but it seems the freshness may have worn off somewhat and fans flocked to the exits after Sarr's third for Tottenham.
The break in the team's winning streak at home won't be a major worry for David Moyes, but their attacking woes will be.
The forward and Thierno Barry - the Toffees' pair of recognised centre-forwards - had netted only a single goal combined before the weekend's match.
And they were unable to add to that poor record on a damp and dreary evening on Merseyside.
Beto's form was epitomized when he opted to bicycle-kick a golden opportunity from near the goal, despite having space and opportunity to readjust for a more conventional shot that would have tested 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 Everton's attacks came through the winger and Ndiaye, who tirelessly whipped dangerous crosses into the Spurs penalty area.
Everton: The team travel to Sunderland in the Premier League on early November at 20:00 GMT.
Tottenham: Spurs travel to Newcastle's ground on late October for their next round Carabao Cup tie against the Magpies.
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