Micky van de Ven became the first backline player to score twice in a Premier League match for Tottenham since Jan Vertonghen in March 2013.
At a glance
Micky Van de Ven nods home twice in opening period
The home side's Jake O'Brien has equalizer ruled after video check
Pape Matar Sarr nets final goal in dying moments
The defeat is the Merseyside club's first at their new ground
Micky van de Ven found the net twice as Spurs extended their winning streak in away games in the Premier League and handed the Toffees their first defeat at Hill Dickinson Stadium.
The Toffees were unbeaten in their last five matches at their recently opened stadium at Bramley-Moore Dock, having left Goodison Park behind after 133 years at the end of the previous campaign.
But Spurs captain the Dutchman pounced on two occasions from corners in the first half to place their impressive record at risk, and they were failed to turn it around before Pape Matar Sarr - assisted by former Toffees attacker the Brazilian - headed home a final goal to put the game to bed.
Jake O'Brien thought he had equalized the scores a short while after the defender's opener, glancing a bullet header past Guglielmo Vicario from an set-piece.
Beto, in the meantime, failed to convert a handful of chances, while the Italian goalkeeper made a series of fine saves to guarantee Tottenham went five matches on the road unbeaten in the English top flight.
But the video assistant referee stepped in, advising the match official to review the goal on the pitchside monitor and the referee ruled that Jack Grealish and the attacker were in offside positions and had impeded with the Spurs keeper.
This victory elevates Thomas Frank's team up to third in the Premier League, a handful of points off north London rivals and leaders the Gunners.
The Merseyside club stay in 14th position and are half a dozen points ahead of the drop zone.
Following a drab midweek performance in the European competition against Monaco, where they mustered a only two shots on target, the manager chose the French striker up front instead of ex-Everton forward Richarlison.
The summer signing was unable to influence the game before being substituted for Richarlison after an hour on the pitch, but it turns out he wasn't needed in the end.
Defender the Dutchman converted into an attacking force to be feared.
Spurs' first goal came from a cleverly executed set-piece play, with the midfielder directing the ball back across the six-yard box for an unmarked the defender to score.
And Ven de Ven - filling in as captain in the Argentine's absence - notched his second in first-half stoppage time as he rose above Jordan Pickford to nod in.
Van de Ven was equally impressive in defence, partnering well with his defensive colleague as the duo handled excellently with the numerous deliveries thrown their way by the opposition.
The club's new home had been a happy hunting ground, but it appears the novelty may have faded somewhat and fans headed to the exits after the midfielder's third for Spurs.
The end in the team's unbeaten home run won't be a major worry for David Moyes, but their attacking woes are.
Beto and the other striker - the Everton's pair of recognised centre-forwards - had scored just one league goal combined before the weekend's match.
And they were unable to add to that miserable record on a damp and dreary evening on Merseyside.
Beto's form was summed up when he opted to bicycle-kick a clear chance from near the goal, despite having space and opportunity to readjust for a simpler shot that would have challenged Vicario.
Barry, as well, was unable to influence the game after replacing the starter in the 65th minute.
It wasn't for a shortage of opportunity. The majority of Everton's attacks came through Grealish and Ndiaye, who relentlessly delivered dangerous crosses into the Spurs box.
The Toffees: The team head to Sunderland in the Premier League on Monday, 3 November at 8 PM.
Spurs: The London club travel to St James' Park on late October for their fourth-round Carabao Cup tie against Newcastle.
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