The Dutch defender emerged as the initial defender to net twice in a English top-flight match for Spurs since Jan Vertonghen in March 2013.
Match summary
The Spurs defender heads into the net two times in first half
The home side's defender has goal disallowed 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 scored on two occasions as Spurs prolonged their winning streak on the road in the top division and delivered the Toffees their first defeat at Hill Dickinson Stadium.
The Toffees were undefeated in their last five matches at their recently opened stadium at the dockside location, having left Goodison Park behind after 133 years at the conclusion of the previous campaign.
But Spurs captain the Dutchman struck on two occasions from set-pieces in the opening 45 minutes to put their impressive record at risk, and they were failed to mount a comeback before Pape Matar Sarr - assisted by former Toffees attacker the Brazilian - nodded in a final goal to put the game to bed.
Jake O'Brien believed he had levelled the tally five minutes after the defender's first goal, glancing a powerful header past Guglielmo Vicario from an set-piece.
Beto, in the meantime, missed a number of chances, while the Italian goalkeeper made a series of excellent stops to ensure the London club went five matches on the road unbeaten in the English top flight.
But the VAR intervened, advising Craig Pawson to review the score 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.
The win elevates the manager's team up to third in the division, a handful of points off local foes and table-toppers Arsenal.
Everton stay in 14th position and are six points ahead of the drop zone.
After a uninspiring weekday performance in the European competition against Monaco, where they mustered a only attempts on target, the manager chose the French striker in attack instead of ex-Everton forward the Brazilian.
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 proved unnecessary in the end.
Defender Van de Ven converted into an attacking force to be feared.
Spurs' first goal came from a well-worked corner routine, with the midfielder heading the ball toward the face of goal for an free Van de Ven to score.
And the captain - stepping up as captain in the Argentine's unavailability - added his next goal in first-half stoppage time as he outjumped Jordan Pickford to nod in.
The Dutchman was equally impressive in defence, combining effectively with Kevin Danso as the pair dealt superbly with the 34 crosses sent their way by the opposition.
The club's recent stadium had been a fortress, but it seems the freshness may have worn off slightly and fans flocked to the gates after the midfielder's third for Tottenham.
The end in Everton's winning streak at home won't be a massive worry for the manager, but their attacking woes will be.
Beto and the other striker - the Toffees' pair of recognised centre-forwards - had scored just one league goal between them before Sunday's match.
And they were failed to contribute to that poor tally on a damp and dreary night on Merseyside.
The striker's performance was summed up when he opted to overhead kick a clear chance from close range, even with having room and time to readjust for a more conventional attempt that would have tested Vicario.
The substitute, too, was unable to influence the game after replacing the starter in the second half.
It wasn't for a shortage of chance. Most of Everton's attacks came through Grealish and the forward, who tirelessly delivered dangerous balls into the Tottenham penalty area.
Everton: Moyes' men travel to the Stadium of Light in the Premier League on Monday, 3 November at 8 PM.
Tottenham: Spurs go to Newcastle's ground on Wednesday, 29 October for their fourth-round League Cup tie against the Magpies.
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