The Bosnian acted furiously when he was taken off during the Blues' 2-0 defeat at the Allianz Arena, but he insisted that his relationship with Mancini is fine.
"I know my reaction 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 okay and that we have to be positive for the next game."
The incident with Dzeko was overshadowed on the night by Carlos Tevez's alleged refusal to come off the substitutes bench, with the Argentinan striker suspended for two weeks by the club because of his actions.