Spurs knocked out the holders through goals from Timo Werner and Pape Sarr. Matheus Nunes struck for City, but the visitors couldn't stage a fight-back in the second-half.
Afterwards, Spurs midfielder Dejan Kulusevski said: "It is important to win and go through the cup. When you play a great time like City you always want to win - I am proud of the boys.
"I always love this game, playing the best teams in the world, I want to be one of the best player in the world.
"We have to keep going and win every game. We have to keep focused."
Werner also said: "It is always hard to play against a team like Man City.
"It is good to score, especially when it is so early in the game. The last few weeks were tough, but you can't stop you always have to go on."