Julian Schieber netted the winner in the 57th minute, condemning City to their third successive away defeat in Europe’s premier club competition this term.
Mancini’s men finished their Champions League campaign without a single victory and bottom of Group D, and the City manager admitted his disappointment to miss out on third spot – and a Europa League place.
“The big problem is that to win games you need to score,” said Mancini.
“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.”
He added: “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.”
Mancini believes City can learn lessons from Group D winners Dortmund, who have shown staggering progress in the Champions League over the past season.
“Last year Borussia Dortmund went out in the first group and were fourth in that group,” he added.
“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.”