The 24-year-old has signed a four-and-a-half year contract with the Eastlands outfit, who are said to be paying Wolfsburg ¦pound;27million for the striker's services.
Emmanuel Adebayor and Roque Santa Cruz now look set to leave this month, but the giant Bosnian is happy to get the chance to settle in, and told mcfc.co.uk: "Manchester City were my first choice and only choice."
He added: "I was always attracted most to joining them. I have been linked with them for a long time and have watched them a lot on television.
"They have been playing well this season and now the speculation is over, I just want to do my best."
Dzeko did not complete his transfer in time to be considered for Sunday's FA Cup tie at Leicester and could instead make his debut in the Premier League clash with Wolves at Eastlands on January 15.
And, after initially declaring Chelsea would run away with the title, then stating this week that City were fifth favourites for the crown, it appears Mancini gave away his real feelings to Dzeko when they met up earlier this week.
"City are playing for first place in the Premier League and that is what I want," confirmed Dzeko.
"I am excited to play alongside players such as Carlos Tevez and Yaya Toure, whom I have only seen on the television, and to be in the same team as players I have played against such as Jerome Boateng and Vincent Kompany. They will help me achieve my goal to be better in the future.
"I want to show the fans why the club paid the money for me. Everyone expects goals and I want to score them but most of all I want to contribute to a successful team."