Enzo Fernández put City in front with a free-kick in the ninth minute and Brobbey answered three minutes later, the first of his three goals. Rayan Cherki restored the lead on 29 minutes, Brobbey levelled again on 33 and Antoine Semenyo made it 3-2 on 43.
Semenyo struck again on 57, Brobbey completed his hat-trick two minutes after that, and Erling Haaland's 81st-minute goal, given after a VAR check for offside, settled it.
Brobbey, who had not scored in the league before, became the first player to score a hat-trick for a losing Premier League side since Roque Santa Cruz in 2007. Haaland's goal meant he has now scored against all 25 opponents he has faced in the competition, and City lead the table by three points after Arsenal lost at Brighton on Saturday.
Sunderland head coach Régis Le Bris told Sky Sports:
“No compromise with our identity. I think we are really proud to play our football. We went toe-to-toe with one of the best sides in Europe today.
“We created many chances and any small error or mistake gets punished at this level. This is the reality of the Premier League and we have to learn.
“The team is really resilient and the belief is high about our qualities and the way we can hurt the opponents. We are a bit frustrated because we came here with the belief to win.”
On Brobbey's hat-trick, Le Bris said:
“It's really positive for a striker, when you don't score it's still possible to tell the story differently but for us it is really clear that he is a good striker and he's getting better game after game. It's just a question of time.”
On Elliot Anderson's potential handball in the penalty area, he said:
“It was hard to see that as not a handball. It is what it is and we have to accept the decision. It's hard to comment.”
Manchester City manager Enzo Maresca, also speaking to Sky Sports, said:
“Sunderland are a very good team. They're strong and dangerous, but again we found a way to win the game.”
On it being a bonkers game to watch, he said:
“Not for us! In the press conference before the game I said I expect a very tough game. Then we also conceded two or three goals but again. We have learnt a lot in these eight weeks and this is good.”
On the high score line, he added:
“Too many goals, I prefer 1-0 it's much better.”