City were largely outplayed by the German champions and only managed to salvage a point through Mario Balotelli's late penalty.
However, Mancini believes that the point could prove crucial should City make it out of their group.
"We didn't play well," Mancini said. "We didn't deserve to take this point but in the end it could be important. Borussia Dortmund played better than us and at this moment they are a better side.
"We need to run and fight for every ball. It's not enough to have just a good quality. I knew before the game it was going to be difficult as Dortmund have more experience in the Champions League than us."
City are third in Group D, three points behind Dortmund and five away from current leaders Real Madrid.