The Bosnian complained bitterly after being replaced by Nigel de Jong 10 minutes into the second-half of the 2-0 defeat to Bayern.
He was still complaining as he took his place on the bench and continued his moans as the match went on.
Mancini was not impressed and stated afterwards it was the "last time any player behaved like that" after being substituted.
Dzeko refused to speak to reporters as he left the stadium.
However, having had chance to consider his actions, he has now said sorry to Mancini and his team-mates.
"I know my reaction (to being taken off) was bad," he said.
"I have spoken to the guys and to the coach as well.
"I have apologised for the reaction and Roberto has accepted it and said that everything is OK and that we have to be positive for the next game."
Dzeko explained that he had been extra motivated to win the game because of the time he spent in Germany with Wolfsburg.
Unfortunately, things did not go to plan for City.
"I was unhappy because we were 2-0 down and I wanted to win the game," he said.
"It was something special for me to go back to Germany where I played for a long time. I wanted to do well and wanted the team to do well. Things didn't go well for us. That is why I was extra frustrated."