City held on for a 1-0 win at Old Trafford despite playing much of the match with 10 men after Phil Foden was sent off in the first half.
Haaland scored the decisive goal in the 60th minute, although the goal came after a lengthy VAR review. The Premier League refereeing body Pro Ref later admitted there had been an "error of judgement" in the decision.
The strike was Haaland's ninth goal in Premier League Manchester derbies, taking him past both Sergio Aguero and Wayne Rooney for the most goals in the fixture.
City's task became even harder when Foden was dismissed in the 23rd minute after reacting to an apparent kick from United captain Bruno Fernandes.
United pushed forward after the red card and came close to equalising, with Marcus Rashford and Kobbie Mainoo both hitting the post.
Then, deep into stoppage time, Fernandes found Bryan Mbeumo with a clever pass into the City penalty area. Mbeumo created space for a shot, but Donnarumma reacted quickly to push the effort away and preserve the three points.
"Everyone just fights like never before and it's amazing to see," Haaland said.
"We have to be proud because this gives motivation. This gives an amazing feeling inside all of us.
"And again, this is the kind of thing we expect from Gigio because he's one of the best goalkeepers in the world. So for him to save this is expected.
"But again, it's the same as me. Before the game, you probably expected me to score a goal and I did. I've done my job. Everyone's done their job. Gigio managed to save us in the 95th minute. He's a monster."
Haaland also explained the instinct behind his difficult finish, which came after Josko Gvardiol's cross took a deflection off Patrick Dorgu.
Despite being off balance, the Norway striker adjusted brilliantly and fired the ball into the roof of the net.
"It's just pure instinct, of what I feel is the right thing to do," Haaland said.
"In the end, there's no solution to this or that. How to finish, how to do this, how to do that, it's just about trying to get the ball in the back of the net and shooting good enough so the goalkeeper cannot save it.
"That's just what I'm trying to think of."
City now turn their attention to the EFL Cup, when they host Norwich City in the third round at the Etihad Stadium on Thursday.
Manchester United surrendered their advantage to draw 1-1 with Tottenham Hotspur at Old Trafford on Saturday evening, missing the opportunity to claim their second Premier League win of the campaign.
Chelsea bounced back from their defeat before the international break with a convincing 5-1 Premier League victory over Bournemouth at Stamford Bridge, as Jordan Henderson scored his first goals for the club.
Arsenal came from behind to secure a hard-fought 2-1 victory over Leeds United at the Emirates Stadium on Saturday, showing resilience to claim maximum points after falling behind in the second half.
Cole Palmer has signed a one-year contract extension with Chelsea, keeping the England international at Stamford Bridge until 2034.
