It was the 27-year-old's 10th appearance in England's top flight following his £27million transfer from Wolfsburg in January.
City are now four points clear of fifth-placed Tottenham and Dzeko wants them to maintain the momentum when they entertain struggling West Ham on Sunday.
He said: "We have five more league games to play and we have to win the next one as it is at home.
"It was a very tough game against Blackburn, very difficult with too many long balls. But we won and that's important.
"Sometimes you don't have to play from the beginning to score. Sometimes you can't score from a lot of chances and sometimes you get one chance and you score. That's football.
"I want to thank the fans for giving me big support. Even though I wasn't scoring the fans were behind me."