Micky van de Ven emerged as the first defender to score twice in a Premier League game for Spurs since the Belgian defender in spring 2013.
Key points
Micky Van de Ven heads home two times in first half
Everton's Jake O'Brien has goal disallowed after VAR review
Senegalese midfielder scores third in 89th minute
The defeat is Everton's initial at their new ground
Micky van de Ven scored on two occasions as Spurs extended their winning streak on the road in the Premier League and handed Everton their first defeat at Hill Dickinson Stadium.
The Toffees were unbeaten 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 conclusion of the previous campaign.
But Tottenham's leader the Dutchman struck on two occasions from set-pieces in the opening 45 minutes to place their impressive record at jeopardy, and they were unable to turn it around before the midfielder - set up by former Toffees attacker the Brazilian - nodded in a final goal to put the game to bed.
The Everton defender believed he had levelled the scores five minutes after the defender's first goal, directing a bullet header past Guglielmo Vicario from an set-piece.
The Portuguese striker, meanwhile, missed a handful of opportunities, while the Italian goalkeeper made a succession of fine stops to guarantee Tottenham went five matches on the road without defeat in the Premier League.
But the VAR stepped in, instructing Craig Pawson to review the goal on the pitchside monitor and the official ruled that Jack Grealish and Iliman Ndiaye were in offside positions and had interfered with the Tottenham goalkeeper.
This victory moves Thomas Frank's team up to third in the Premier League, five points off north London rivals and table-toppers Arsenal.
The Merseyside club stay 14th and are six points ahead of the relegation zone.
Following a uninspiring weekday display in the European competition against Monaco, where they managed a only two shots on target, the manager chose Randal Kolo Muani up front instead of ex-Everton forward Richarlison.
The new arrival was unable to influence the game before being replaced for the substitute after 60 minutes on the pitch, but it turns out he proved unnecessary in the end.
Centre-back Van de Ven converted into an offensive threat to be feared.
Spurs' first goal came from a cleverly executed set-piece play, with Rodrigo Bentancur heading the ball back across the face of goal for an free Van de Ven to head home.
And the captain - stepping up as skipper in the Argentine's unavailability - notched his next goal in first-half stoppage time as he rose above the England keeper to nod in.
The Dutchman was just as outstanding in defensive duties, combining well with his defensive colleague as the pair handled excellently with the 34 crosses thrown their way by the opposition.
Everton's new home had been a fortress, but it appears the freshness may have faded slightly and fans flocked to the exits after Sarr's final goal for Spurs.
The end in the team's unbeaten home run will not be a major concern for David Moyes, but their attacking woes will be.
Beto and the other striker - the Everton's pair of recognised strikers - had scored just one league goal between them before Sunday's match.
And they were unable to contribute to that poor tally on a wet and gloomy evening on in Liverpool.
Beto's performance was epitomized when he chose to bicycle-kick a golden opportunity from near the goal, despite having room and opportunity to readjust for a more conventional shot that would have challenged Vicario.
The substitute, too, was unable to influence the game after coming on for Beto in the 65th minute.
It wasn't for a shortage of opportunity. The majority of Everton's offensive moves came through Grealish and the forward, who relentlessly whipped threatening balls into the Tottenham box.
Everton: The team head to the Stadium of Light in the top flight on Monday, 3 November at 8 PM.
Tottenham: The London club travel to St James' Park on late October for their next round Carabao Cup tie against the Magpies.