United overturned a 2-1 deficit from the first leg at City, leaving Mancini to concentrate on the Premier League and FA Cup in his first season in charge of the big-spending club.
"It's disappointing for the players, I feel very sorry for them and the supporters because I would like to have reached the final for them," Mancini told a news conference.
"Now we have got some players back - (Kolo) Toure and (Emmanuel) Adebayor - and (Joleon) Lescott is coming back, so because of this I think it's possible we can finish in the top four.
"I think that you when you lose against United, it is important that you can learn. It's too easy to improve when you win, you must do it when you lose. We must work in the next two or three months, we can finish in the top four and we are still in the FA Cup.
"It is important the players go away with their heads held high, I think the players played these two games very well."
City, who have not competed in a major final since losing the 1981 FA Cup Final to Tottenham and not won a trophy since the 1976 League Cup, face a home FA Cup fifth-round tie against Stoke City.
But Mancini's plans appear certain not to include Robinho after the unsettled Brazilian striker was left out of the 18-man squad against United.
"I spoke with Robinho yesterday and it is possible he can play for Santos or another team," said Mancini of the proposed loan departure.
"Because of this we preferred not to risk him."
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