The former Wolfsburg striker was out of luck on Tuesday night during the 0-0 away draw with Aris Salonika in the Europa League.
Dzeko almost opened the scoring in the first leg of the last-32 tie in the 16th minute when he got on the end of a driven cross from Aleksandar Kolarov, but saw his close-range shot gathered by Michalis Sifakis at the second attempt.
And the Bosnian again came close in the 64th minute when, following a cross from Gareth Barry, he got in a towering header but Sifakis somehow managed to tip it over the top of the bar.
"I expect more for Dzeko - he needs to improve. He is a good player," said Mancini. "I think he has the same difficulty as (Fernando) Torres at Chelsea because when you arrive in January you don't know your team-mates, it is not easy.
"He can change in every game. It is important that he improves very quickly. We should cross the ball more to him."