Greek minnows Aris Thessalonika were never in the hunt once Dzeko had made his double contribution inside the first 12 minutes - they eventually lost 3-0.
Yaya Toure's deflected effort brought City a third before the end to confirm a meeting with Dynamo Kiev.
Dzeko told ESPN: "I am very happy because I scored two goals in one game for the first time for Manchester City and we are in the next round.
"We scored two goals in the first 10 minutes and this was very important for this game."
The Serbia international believes he is settling in to life at Eastlands but he was disappointed he did not score a hat-trick.
"I can play much better.
"I am not happy with the last chance of the first half, I have to work."
Mancini had started with Dzeko, Mario Balotelli and Carlos Tevez up front of which he said: "The first half (they did) very well, it was the first time they need to play again but second half we were so tired."
The Italian admitted it took until the third goal for him to relax.
"After that goal yes, before no.
"In football never say never. If they scored one goal it would reopen the game."
As for the last-16 game against Kiev, Mancini said: "I have played against them a lot of times, they are a big team.
"They always play in the Champions League, they have lots of experience, it will be a very difficult game."