Micky van de Ven emerged as the initial backline player to score twice in a Premier League match for Tottenham since Jan Vertonghen in March 2013.
Key points
Micky Van de Ven heads home two times in opening period
The home side's Jake O'Brien has goal disallowed after video check
Pape Matar Sarr nets third in dying moments
The defeat is the Merseyside club's first at their new ground
The Tottenham defender found the net twice as Spurs prolonged their winning streak 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 the dockside location, having departed their historic home behind after more than a century at the end of last season.
But Spurs captain Van de Ven pounced twice from set-pieces in the first half to put their impressive record at jeopardy, and they were failed to turn it around before Pape Matar Sarr - assisted by former Toffees attacker Richarlison - headed home a final goal to put the game to bed.
Jake O'Brien believed he had equalized the scores a short while after Van de Ven's opener, directing a powerful header past Guglielmo Vicario from an set-piece.
Beto, in the meantime, failed to convert a handful of opportunities, while the Italian goalkeeper made a succession of fine stops to ensure Tottenham went five away games unbeaten in the English top flight.
But the video assistant referee stepped in, advising Craig Pawson to check the score on the screen and the official decided that the winger and the attacker were in offside positions and had interfered with the Tottenham goalkeeper.
The win elevates the manager's side up to third in the division, five points off local foes and leaders Arsenal.
Everton remain 14th and are six points ahead of the drop zone.
Following a uninspiring weekday display in the European competition against Monaco, where they managed a only two shots on goal, the manager opted for the French striker in attack instead of former Toffees player Richarlison.
The summer signing was failed to influence the game before being substituted for Richarlison after 60 minutes on the field, but it emerged he proved unnecessary in the final outcome.
Centre-back Van de Ven transformed into an offensive threat to be feared.
Spurs' first goal came from a cleverly executed corner routine, with Rodrigo Bentancur directing the ball back across the face of goal for an free the defender to head home.
And Ven de Ven - filling in as skipper in the Argentine's unavailability - added his second in added time as he outjumped the England keeper to head home.
The Dutchman was just as outstanding in defence, partnering well with his defensive colleague as the duo dealt superbly with the 34 crosses thrown their way by Everton.
Everton's new home had been a fortress, but it appears the freshness may have worn off slightly and fans flocked to the exits after Sarr's third for Tottenham.
The break in the team's winning streak at home will not be a massive worry for the manager, but their striker situation are.
Beto and the other striker - the Everton's two recognised strikers - had scored just a single goal between them before Sunday's match.
And they were failed to add to that poor tally on a damp and dreary night on Merseyside.
Beto's form was epitomized when he chose to bicycle-kick a golden opportunity from near the goal, despite having room and time to readjust for a more conventional shot that would have challenged the keeper.
The substitute, as well, was unable to make an impact after replacing Beto in the 65th minute.
It wasn't for a lack of chance. The majority of Everton's attacks came through Grealish and the forward, who tirelessly whipped dangerous crosses into the Tottenham box.
The Toffees: The team travel to Sunderland in the Premier League on early November at 8 PM.
Tottenham: Spurs go to St James' Park on late October for their next round League Cup tie against Newcastle.
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