The Bosnian fired home the only goal of the game in the 75th minute, just three minutes after emerging from the substitutes' bench.
And he told Sky Sports 1: "I'm very happy that this goal helped my team to win the game. Sometimes it's difficult, I'm not the first who didn't make the best start but I'm happy because I scored today and helped the team.''
Dzeko's drought has been focused on parallel with that of Chelsea's £50 million man Fernando Torres, who opened his Blues account against West Ham on Saturday.
And Dzeko said: "It's football, Torres scored a lot of goals at Liverpool and changed club, I'm happy for him too because I know what the pressure is. Now we have to look at the future - today was very good, it is a new beginning for me and for City.''
Manager Roberto Mancini added: "I'm very happy for Edin because he deserves to score, he is a good player and a good man. In the first half we played a fantastic game, we should have scored two goals minimum. In the second half the game changed, Blackburn played very strong and hard.''
Blackburn boss Steve Kean said: "We're bitterly disappointed, we gave it a right go. After the first 15 minutes we got to grips in midfield and were in the ascendancy before half-time. Even in the second half we were excellent.''
"There was contact, Jason was saying he wanted to go through and finish,'' he said. "We also thought we had a late shout with Niko Kalinic. On the goal, (Mario) Balotelli was right in front of Paul Robinson so I'm saying that was offside.''
Nevertheless, Kean remains confident of top-flight survival and said: "We feel as a group if we play like that against Bolton on Saturday we can get a win, never mind a draw.
"We're gutted we didn't get anything tonight but if we play like that, we've got a great chance.''