The England number one produced several outstanding saves before Mario Balotelli struck a late penalty. Dortmund keeper Roman Weidenfeller was also kept busy in an end-to-end encounter, but the visitors had looked on course for victory after Marco Reus pounced on a Jack Rodwell mistake after 62 minutes.
Mancini said: "Joe Hart saved us from an incredible defeat because he saved everything and we should say thank you to him."
Asked whether it was the best goalkeeping display he had ever seen, Mancini said: "I don't know this. He did a fantastic performance, he was fantastic tonight. I don't know if it is the best performance in football history, but he did very well."
The half-time score could easily have been 3-3, rather than 0-0, as both teams looked to attack, but Mancini was disappointed by his side's efforts. The result leaves them with just one point from their opening two fixtures in the competition and facing an uphill fight with back-to-back fixtures against Ajax to come next.
City have also still not kept a clean sheet all season and, despite Hart's brilliance, their defence is a concern.
"We didn't play well. We didn't deserve to take this point but in the end it could be important," City manager Roberto Mancini told Sky Sports.
"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.
"I think that Joe Hart saved us because he did very, very well. I know the problem and I will solve it very quickly. If we don't improve it will be difficult (to qualify) but this point could be important."
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."