The Bosnian went down under Laurent Koscielny's 10th-minute challenge and, although he missed the resulting penalty, City made the resulting man-advantage tell.
James Milner was first on the scoresheet - blasting home sweetly off the far post - and Dzeko made amends for his spot-kick blunder by tapping home just after the half-hour mark.
Referee Mike Dean levelled the teams up late on when Vincent Kompany lunged in on Jack Wilshere, but the champions saw out the game to stay seven points behind arch-rivals United.
Man-of-the-match Dzeko told Sky Sports afterwards: "It was a penalty. I didn't score but I knew I was going to score afterwards because I told myself 'Ok you missed the penalty, now you have to work hard and try to score from another chance.'
"It's very important for my confidence to know you're going to play the next game. I think I've started the last four games and I think I've scored four goals so I'm happy.
"The most important thing is that the team is winning the games and that we are still there fighting for first place."
Milner added: "It's a tough place to come and we haven't had much success here in recent years. We wanted to put that right and we got three points.
"I saw a little pocket to run into and it was a nice weight of pass and I thought I'd have a go. Luckily enough, it went in."
On Kompany's red card, the England wide man added: "It's hard to say at that pace. It's a tough job being a referee and it could've gone either way. It's difficult to say without having another look at it."