City had to battle to the result after skipper Vincent Kompany sent off on 10 minutes.
Goalscorer Dzeko remarked: "Vinny is our captain and our leader, our heart, and I hope he will be back soon so we can win games together.
"We showed character with the 10 men over 80 minutes and we did a good job.
"We were fighting all game because we knew what the game meant for us. We won the game and that’s important.
"The next game against Fulham is important and we will try to win that."
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