The Dutch defender emerged as the initial backline player to score twice in a English top-flight match for Spurs since Jan Vertonghen in spring 2013.
At a glance
The Spurs defender nods home two times in first half
The home side's defender has equalizer disallowed after video check
Senegalese midfielder nets third in dying moments
The defeat is the Merseyside club's initial at Hill Dickinson Stadium
Micky van de Ven found the net on two occasions as Spurs extended their unbeaten run in away games in the Premier League and delivered Everton their initial loss at Hill Dickinson Stadium.
The Toffees were unbeaten in their last five matches at their new ground at Bramley-Moore Dock, having left their historic home behind after more than a century at the conclusion of last season.
But Tottenham's leader the Dutchman pounced twice from corners in the opening 45 minutes to place their strong form at jeopardy, and they were unable to mount a comeback before the midfielder - set up by ex-Everton forward Richarlison - headed home a third to put the game to bed.
The Everton defender believed he had levelled the scores a short while after Van de Ven's opener, glancing a bullet header past Guglielmo Vicario from an set-piece.
Beto, meanwhile, failed to convert a number of opportunities, while Vicario made a succession of fine stops to guarantee the London club went five matches on the road without defeat in the English top flight.
But the VAR intervened, advising the match official to check the goal on the pitchside monitor and the official ruled that the winger and Iliman Ndiaye were in offside positions and had interfered with the Tottenham goalkeeper.
This victory moves Thomas Frank's team into third place in the Premier League, a handful of points off local foes and leaders Arsenal.
The Merseyside club remain 14th and sit half a dozen points ahead of the relegation zone.
After a uninspiring weekday display in the European competition against the French side, where they managed a only attempts on target, the manager opted for the French striker in attack instead of ex-Everton forward the Brazilian.
The new arrival was unable to make much of an impact before being substituted for the substitute after an hour on the field, but it emerged he wasn't needed in the final outcome.
Defender the Dutchman converted into an attacking force to be reckoned with.
Tottenham's first goal came from a cleverly executed corner routine, with the midfielder directing the ball toward the face of goal for an unmarked the defender to head home.
And Ven de Ven - filling in as captain in Cristian Romero's unavailability - added his next goal in first-half stoppage time as he rose above the England keeper to head home.
The Dutchman was equally impressive in defence, combining well with his defensive colleague as the duo dealt superbly with the 34 crosses sent their way by Everton.
The club's new home had been a fortress, but it appears the freshness may have worn off slightly and fans flocked to the gates after the midfielder's final goal for Spurs.
The break in Everton's winning streak at home will not be a major concern for David Moyes, but their attacking woes are.
Beto and Thierno Barry - the Everton's pair of established centre-forwards - had scored only a single goal combined before Sunday's outing.
And they were unable to contribute to that poor tally on a damp and dreary night on Merseyside.
Beto's performance was summed up when he opted to bicycle-kick a golden opportunity from close range, despite having room and time to reposition for a simpler shot that would have challenged Vicario.
The substitute, as well, failed to influence the game after coming on for Beto in the second half.
It was not for a lack of opportunity. The majority of Everton's offensive moves came through the winger and the forward, who tirelessly delivered dangerous crosses into the Spurs box.
The Toffees: Moyes' men head to Sunderland in the Premier League on early November at 8 PM.
Spurs: The London club travel to Newcastle's ground on late October for their next round League Cup match against Newcastle.