The German visitors went into the lead in the second half when Marco Reus capitalised on a Jack Rodwell error, and could have increased their lead only for an excellent performance from Joe Hart.
With the Premier League champions staring defeat in the face, Mario Balotelli netted a last-minute penalty to ensure spoils were shared.
Roberto Mancini admitted that Hart’s saves spared his side’s blushes and praised Dortmund.
“I think that Joe saved us. He did very, very well,” the Italian told Sky Sports.
“Today we deserved to concede three or four goals. We didn’t play well. We didn’t play well because Borussia Dortmund played better than us.
“They probably at this moment are a better team than us. If we want to play in the Champions League we have to play better than tonight.
“We have to run and fight. It’s not enough to just pass. The Champions League is another situation – different from the championship. We have to run and fight. It’s not enough to just have quality,” he concluded.
Real Madrid beat Ajax 4-1 in the other game in Group D.