The Premier League champions had to come from 2-1 down to get the new campaign off to a winning start at the Etihad Stadium.
"We were not 100%. We did some mistakes but the big problem was we missed five or six chances when we were 1-0 and then it changed," Mancini told BBC Sport.
"We deserved to win the game. They didn't have any chances in the first half and we had four or five incredible chances."
City's next Premier League fixture is away to Liverpool on Sunday.