Trailing 1-0 to Sporting after the first-leg loss in Lisbon, City were left needing to score four times in the second half of the return fixture at the Etihad Stadium following efforts from Matias Fernandez and Ricky van Wolfswinkel.
The Blues managed three of them, as a Sergio Aguero brace and Mario Balotelli penalty levelled the tie at 3-3, with the Portuguese outfit ahead on away goals.
And as the seconds ticked down, England international Hart sauntered forward for a last-gasp corner-kick and saw his goal-bound header finger-tipped to safety by Rui Patricio.
"I can't believe it, I thought it was in," he said. "It didn't need to get to that, but football is football.
"The fans were fantastic and we didn't give them what they deserved in the first half.
"Hopefully we can take some positives out of the second half and move on for the rest of the season."
Roberto Mancini's men are not back in action until next Wednesday when they entertain a rejuvenated Chelsea, and could be four points behind city rivals Manchester United should the Red Devils defeat Wolverhampton Wanderers.
"We need to play well in every game," Hart added. "There are not many left now. We've cost ourselves with the first half but if we keep rolling now we'll be all right.
"We're one point behind the leaders of the Premier League, we're a good team and we've got nothing to worry about."
