A 0-0 draw against Aris Salonika in Greece would appear to put them in the driving seat for the second leg of their last-32 tie at Eastlands next week.
Mancini said: "This result is better than 1-1. We have a chance because we are at home. It will be a different pitch. It wasn't a good pitch to play football. In the first half Aris got players behind the ball and it was very difficult for us. We could have changed things when (Edin) Dzeko had a chance."
He added: "I think it will be a different game in Manchester. We played well, we played football and controlled the game. Aris stayed behind the ball. It was a fantastic atmosphere."
Former Wolfsburg striker Dzeko should have made the breakthrough only for Michalis Sifakis to save at the second attempt.
Then in the second half the goalkeeper made a superb save from the Bosnian's towering header.
Mancini then chided Dzeko and Shaun Wright-Phillips for not providing the striker with enough good crosses.
He said: "I expect more from Dzeko - he needs to improve. He is a good player.
"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. Shaun should have done better than today."