City agreed to pay German side Wolfsburg £27million for the prolific Bosnian international striker during this January transfer window.
And, following his success in helping them win their first Bundesliga title two years ago, Dzeko is geared up to banishing City's long wait for silverware and in particular help them secure the Premier League title.
However, he accepts that he will need the fans to be patient while he adapts to his new surroundings and the pace of the game in England.
"I have to take it step by step. I know I will have to work hard, I will have to get used to football in England," he told Dnevni Avaz.
"Without doubt City are fighting for first place. Success at the club is very important. That's why I'm here and that's something I will work as hard as I can to help."
On the prospect of working under Roberto Mancini at Eastlands, Dzeko added: "My first impression is great. He (Mancini) was also so happy that the transfer was finally completed. He wanted me the most. I can say that he is a gentleman in every meaning of the word," he continued.
"I'm just so happy that everything is now completed. There were so many crazy and stupid things written in the media, but I'm glad it's finally over.
"I'm still waiting to celebrate. There hasn't been any time at all. Firstly I have to have a good sleep and get ready to play for my new club.
"I was ill for a few days and I've lost some strength. I have to get my energy back and I hope to be as ready as I can for Wolves."