City could not even secure the consolation of a place in the Europa League as they slipped to a 1-0 defeat at the Westfalenstadion in their final Champions League Group D game.
Mancini felt his team gave their best but paid the price for failing to convert chances, a problem he feels is growing.
Mancini said: "The big problem is that to win games you need to score.
"If not, you lose or the game finishes 0-0. At the moment we are not scoring a lot of goals, as we did last year.
"When we get chances we need to score - we need to resolve this problem if we are going to win."
City entered the competition with high hopes but have finished bottom of their group with just three points, the worst showing by an English side in the competition.
Mancini claims he is not embarrassed by the poor return or by the failure to win a place in the Europa League.
He said: "It can be embarrassing if, when you play, you don't play at 100 per cent.
"But when you do play 100 per cent, you can lose.
"We wanted to try to get in the Europa League because that would have been another competition for us to try to win."
Julian Schieber scored the only goal of a subdued game after 57 minutes. It brought a suitably sorry end to City's campaign, which never recovered after they conceded two late goals to lose their opening match at Real Madrid.
They managed three home draws but lost all of their away games.
Mancini feels Dortmund, who suffered a premature exit last season, are a good example for them as they look to improve in future.
He said: "Last year Borussia Dortmund went out in the first group and were fourth in that group.
"This year, for me, they are probably a team that can win the Champions League.
"When you start playing Champions League, if you make some mistakes, like we did in the first two or three games, you don't have time to recover.
"We were 2-1 up in Madrid with four minutes to go, and we were winning 1-0 in Amsterdam and then had two or three chances to score another goal.
"If we make these mistakes, afterwards, you have no time."
City must quickly put the disappointment aside to prepare for Sunday's top-of-the-table Barclays Premier League clash with rivals Manchester United.
Mancini said: "I don't think there will be a problem with morale.
"We didn't lose the Champion League tonight - it's clear that tonight we lost the chance to qualify for the Europa League.
"But I think that after that, it is important that we only think about the derby.
"We have five days to recover our strength, and then I don't think there will be a problem."
City have decided to stay in Germany for an extra night rather than travelling straight home.
Mancini said: "It is better for the players to stay here overnight and recover, otherwise we will arrive in Manchester at five o'clock in the morning."
There was little riding on the game for Dortmund as they had already been confirmed as group winners but coach Jurgen Klopp said he had no thoughts of easing up.
Klopp said: "It was a great evening in the Champions League and I thought the lads played great.
"People said we wouldn't try but that was not the case.
"I had every confidence and we had fun in the job.
"We played real football and had good possession.
"In the second half we moved up a step.
"It was a great goal and it was great the lads could play and have lots of fun.
"We are a difficult opponent to play against."