Dzeko, 25, joined City from Wolfsburg in January for a fee of £27 million but initially failed to reach the heights he had demonstrated at the Bundesliga club, scoring just twice in 15 Premier League appearances.
This campaign, however, the Bosnia international has been in inspirational goalscoring form, notching seven times in four matches for Mancini's side, including four netted in the 5-1 win over Tottenham on Sunday.
Dzeko said: "I am happy that he kept faith with me because I think that's the most important thing in football - that the coach and the players around you believe in you.
"The beginning was hard for me, but I wanted to have a full pre-season with the team. Now I have had that and I know the league better. I feel much stronger than last season.
"Thinking about the Golden Boot is too far ahead. I don't think about that. I think every striker's dream is that but it's not the priority for me."