Man City made it four wins from as many games in the league and broke Man United's 100% start to the season, but United boss Jose Mourinho was left frustrated with referee Mark Clattenburg.
The Portuguese felt a second-half sliding challenge from Bravo on Wayne Rooney in the box warranted a penalty and a red card for the Chilean.
Mourinho said: "We are very disappointed with the result, but I am very disappointed with two decisions of Mark (Clattenburg).
"It was a penalty and a red card for (Claudio) Bravo. For sure outside the box it is a free-kick and a red card. Inside the box it has to be the same, but obviously even the best referees make mistakes."
On the performance of his side, Mourinho admitted he would not have chosen the same starting line-up given the benefit of hindsight.
He added: "I had two or three players in the first half that, if I know what is going to happen, I don't play them. But this is football and sometimes players disappoint managers.
"It's my fault because I'm the manager and it's always my fault because it's my choice.
"I think in the first half they were much better than us. We started the game bad, with some players really below their normal level in terms of their concentration.
"The second half was completely different. We were in control, we were much more dangerous and we had chances to equalise and who knows what would have happened."