The Dutch winger, signed from PSV Eindhoven in the January transfer window, chested the ball and volleyed into the bottom-right corner from 18 yards to give Spurs a 63rd-minute lead.
Son Heung-min doubled Spurs' advantage in the 71st minute, firing low past Ederson.
In the first half, Sergio Aguero's shot was diverted on to the post by Hugo Lloris.
Then the Argentinian won a penalty after being fouled by Serge Aurier, following a consultation with the Video Assistant Referee. Lloris saved Ilkay Gundogan's spot-kick.
City were reduced to 10 men on the hour mark when Oleksandr Zinchenko was shown a second yellow card for a foul on Harry Winks, and Spurs took advantage.
Victory means Spurs trail fourth-placed Chelsea by four points. Man City remain second with 51 points, 22 points behind leaders Liverpool.
Spurs: 16 Feb v Aston Villa (A)
Man City: 9 Feb v West Ham (H)