Micky van de Ven became the first backline player to net two goals in a Premier League match for Spurs since Jan Vertonghen in March 2013.
Match summary
Micky Van de Ven nods home twice in first half
Everton'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 Hill Dickinson Stadium
The Tottenham defender scored twice as Spurs prolonged their winning streak on the road in the Premier League and handed the Toffees their initial loss at Hill Dickinson Stadium.
Everton were undefeated in their five previous fixtures at their recently opened stadium at Bramley-Moore Dock, having departed Goodison Park behind after more than a century at the conclusion of last season.
But Spurs captain Van de Ven pounced twice from corners in the first half to put their impressive record at risk, and they were failed to turn it around before Pape Matar Sarr - set up by ex-Everton forward the Brazilian - headed home a final goal to seal the victory.
Jake O'Brien thought he had levelled the tally a short while after Van de Ven's opener, glancing a powerful header past the Spurs keeper from an set-piece.
Beto, in the meantime, failed to convert a handful of chances, while the Italian goalkeeper made a succession of fine stops to guarantee Tottenham went five matches on the road unbeaten in the Premier League.
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 Iliman Ndiaye were in offside positions and had impeded with the Spurs keeper.
This victory moves the manager's side into third place in the division, a handful of points off local foes and table-toppers Arsenal.
Everton remain in 14th position and sit six points ahead of the drop zone.
After a uninspiring weekday display in the Champions League against the French side, where they managed a mere two shots on goal, Frank opted for Randal Kolo Muani in attack instead of former Toffees player the Brazilian.
The summer signing was failed to influence the game before being substituted for the substitute after 60 minutes on the pitch, but it turns out he proved unnecessary in the final outcome.
Centre-back the Dutchman converted into an offensive threat to be feared.
Spurs' first goal came from a cleverly executed set-piece play, with Rodrigo Bentancur directing the ball back across the face of goal for an free Van de Ven to score.
And Ven de Ven - filling in as captain in the Argentine's unavailability - notched his next goal in first-half stoppage time as he rose above the England keeper to head home.
The Dutchman was just as outstanding in defence, partnering well with his defensive colleague as the pair handled superbly with the 34 crosses thrown their way by Everton.
The club's new home had been a happy hunting ground, but it appears the novelty may have worn off slightly and supporters flocked to the gates after Sarr's final goal for Spurs.
The end in Everton's unbeaten home run won't be a massive concern for the manager, but their attacking woes are.
The forward and Thierno Barry - the Toffees' pair of recognised centre-forwards - had netted only one league goal between them before Sunday's outing.
And they were failed to add to that poor record on a damp and dreary evening on in Liverpool.
Beto's form was summed up when he opted to bicycle-kick a golden opportunity from near the goal, even with having space and opportunity to reposition for a simpler shot that would have tested Vicario.
The substitute, as well, failed to influence the game after coming on for Beto in the 65th minute.
It was not for a shortage of chance. The majority of Everton's offensive moves came through the winger and Ndiaye, who tirelessly delivered threatening crosses into the Tottenham penalty area.
The Toffees: Moyes' men 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 Carabao Cup match against Newcastle.