Mancini claimed at the time of Dzeko's arrival from Wolfsburg in January that the striker could "make the difference" in their bid for silverware.
But that has not looked the case so far and he spurned a couple of glorious chances to hand City a first-leg advantage against the Greek outfit in their 1-1 draw on Tuesday.
Mancini feels Dzeko is struggling to adapt to his new home and compared his situation to that of Chelsea new boy Fernando Torres.
"I expect more from Dzeko - he needs to improve," Mancini said. "He is a good player.
I think he has the same difficulty as 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."
Mancini also questioned the role of Shaun Wright-Phillips, who did not impress in a rare start. The Italian said: "We should cross the ball more to him. Shaun should have done better."