The Dutch defender became the first backline player to net two goals in a Premier League game for Tottenham since Jan Vertonghen in March 2013.
Match summary
The Spurs defender nods home twice in first half
Everton's Jake O'Brien has equalizer disallowed after video check
Pape Matar Sarr scores third in dying moments
This loss is Everton's first at Hill Dickinson Stadium
The Tottenham defender scored twice as Tottenham prolonged their winning streak in away games in the top division and delivered Everton their first defeat at their new home.
The Toffees were undefeated in their five previous fixtures at their new ground at the dockside location, having left their historic home behind after 133 years at the end of the previous campaign.
But Spurs captain the Dutchman struck twice from corners 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 Richarlison - headed home a final goal to seal the victory.
Jake O'Brien thought he had equalized the scores five minutes after the defender's opener, directing a bullet header past the Spurs keeper from an Everton corner.
The Portuguese striker, in the meantime, missed a number of opportunities, while the Italian goalkeeper made a succession of fine stops to guarantee the London club went five away games without defeat in the Premier League.
But the video assistant referee intervened, advising the match official to check the score on the pitchside monitor and the official ruled that Jack Grealish and Iliman Ndiaye were in an illegal position and had interfered with the Tottenham goalkeeper.
This victory elevates the manager's side into third place in the division, five points off local foes and table-toppers Arsenal.
Everton remain 14th and sit half a dozen points clear of the drop zone.
Following a drab weekday display in the Champions League against Monaco, where they mustered a mere two shots on target, the manager opted for Randal Kolo Muani in attack instead of former Toffees player Richarlison.
The new arrival was failed to make much of an impact before being replaced for the substitute after 60 minutes on the field, but it turns out he wasn't needed in the final outcome.
Defender Van de Ven transformed into an offensive threat to be feared.
Spurs' opening strike came from a cleverly executed corner routine, with the midfielder directing the ball toward the face of goal for an free Van de Ven to score.
And the captain - stepping up as captain in Cristian Romero's unavailability - notched his second in first-half stoppage time as he outjumped the England keeper to head home.
The Dutchman was just as outstanding in defensive duties, combining well with Kevin Danso as the duo handled superbly with the numerous deliveries thrown their way by the opposition.
Everton's new home had been a happy hunting ground, but it seems the novelty may have worn off somewhat and supporters flocked to the gates after the midfielder's third for Tottenham.
The break in Everton's unbeaten home run will not be a major worry for the manager, but their attacking woes will be.
The forward and Thierno Barry - the Everton's pair of recognised centre-forwards - had scored only a single goal combined before Sunday's outing.
And they were unable to add to that miserable record on a wet and gloomy night on Merseyside.
Beto's performance was summed up when he opted to bicycle-kick a golden opportunity from near the goal, despite having space and opportunity to readjust for a simpler attempt that would have challenged the keeper.
Barry, too, was unable to influence the game after coming on for Beto in the 65th minute.
It was not for a shortage of opportunity. Most of Everton's offensive moves came through the winger and the forward, who tirelessly whipped threatening crosses into the Spurs box.
Everton: Moyes' men head to Sunderland in the top flight on early November at 20:00 GMT.
Tottenham: The London club go to St James' Park on Wednesday, 29 October for their next round League Cup match against Newcastle.