Mario Balotelli struck a contentious late penalty to snatch a 1-1 draw in a pulsating Group D clash at the Etihad Stadium.
City have also still not kept a clean sheet all season and, despite Hart's brilliance, their defence is a concern.
Mancini said: "We didn't play well. We conceded a lot of chances to them and I am really disappointed by our performance.
"The Champions League is totally different from the championship. When you have a chance you need to score. If you concede like we concede this evening, it is hard to win.
"In the first half we had three or four incredible chances. If you don't defend, if you don't run like Borussia Dortmund did this evening, it is difficult.
"This is my problem - is my job. I will solve this problem."
City equalised in the last minute through substitute Balotelli after Neven Subotic was harshly judged to have handled at point-blank range as Aguero hooked the ball toward goal.
Mancini said: "It was important - Mario's penalties are like Joe in goal. He did very well.
"I am happy because he scored. Our performance was very, very poor, but at the end of the group this point could be very important."