José Mourinho's men had been completely outplayed for the first hour, but were kept in the game by City's profligacy and Hugo Lloris's first-half penalty save from Ilkay Gundogan.
However, Oleksandr Zinchenko's sending off changed the course of the game as Spurs scored with their first two shots on goal through Bergwijn and Son Heung-min.
Victory lifts Spurs up to fifth, just four points off fourth-placed Chelsea.
A sixth league defeat of the season for City sees Liverpool remain 22 points clear at the top and within touching distance of their first league title for 30 years.