The England international produced a string of outstanding saves to frustrate the German champions until Marco Reus finally broke through, after swooping on a poor pass from Jack Rodwell on 62 minutes.
City looked set for back-to-back defeats to start their Champions League campaign until they were awarded a contentious late penalty, which Mario Balotelli converted.
"We've got to be happy with a point," said Hart.
"I thought Borussia Dortmund were different class tonight. It took a good defensive effort to keep it at one.
"The pass from Rodwell was a bit short and they took the chance.
"They were getting chances with four at the back or even 10 back. They were always going to get chances.
"For them to be the fourth seed just shows what a tough group we are in."
Hart's performance drew praise from England team-mate Wayne Rooney who labelled him the best goalkeeper in the world during the match on his Twitter page.
Rooney wrote: "Have to say Joe Hart has been incredible. For me, best keeper in the world."
The score could easily have been 3-3 at half-time, but was not due to the brilliance of the keepers.
But the England No.1 could do little as Rodwell gifted the ball to Reus with a poor pass just after the hour and the Dortmund attacker burst forward to blast into the net.
And with the game edging ever closer to its conclusion, City kept the pressure on and gained a very fortunate penalty decision when Sergio Aguero hooked the ball towards goal and it struck Nevin Subotic's arm at point-blank range.
Balotelli did not complain and he stepped up to despatch a cool spot-kick into the bottom corner.
With Ajax falling to a 4-1 thrashing at home to Real Madrid, City sit in third place on just one point as they go into a double header against the Dutch side knowing that they must up their game if they are to progress to the knockout stages.
"I think that Joe saved us. He did very, very well," said City boss Roberto Mancini.
"Today we deserved to conceded 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."