Karim Benzema struck the opening goal for the Spanish champions and Real should have had more in a devastating spell but City grew in confidence as the match wore on and Sergio Aguero levelled from the spot after Alvaro Arbeloa was sent off for a second yellow card.
City poured forward in search of a winner against their 10-man opposition in the closing stages but it was not to be and the result leaves the Barclays Premier League champions with just three points in Group D and unable to progress.
"We're very disappointed. We've had a few opportunities in this group when we could have done a lot better," England goalkeeper Hart told Sky Sports 2.
"All the games have cost us. We've lost our two away games and not won at home.
"We led in Amsterdam and led in Madrid. It's been a disappointing campaign.
"We've surprised ourselves in a very, very bad way in this competition."
The visitors went ahead early on as Benzema peeled away from Maicon to turn in a curling deep cross from Angel di Maria at close range, despite City having plenty of players back in defence.
The goal typified what has been a hugely disappointing campaign for City, and manager Roberto Mancini admitted his side had been below-par on too many occasions.
"The first chance they had, they scored. We made a mistake. We were five against two in the box. When you concede like this it is always difficult," the Italian told Sky Sports 2.
"The first half we played so so. We missed one or two chances.
"But we didn't lose tonight [Wednesday], we lost in the first few games - in Madrid we were 2-1 up with five minutes to go; in Amsterdam we were 1-0 ahead and missed some chances to close the game.
"I am disappointed like everyone here. We did some mistakes.
"When we started this group we knew it was difficult. But we are a good team. You can lose against Real Madrid, Borussia Dortmund, but we are disappointed - we wanted to go to the next stage.
"Now it's finished, we can do nothing."
Mancini also dismissed speculation that City may prefer to finish bottom of their pool following their final group match in Dortmund, rather than take third place ahead of Ajax and with it a place in the often maligned Europa League.
"We want to be in the Europa League, if it is possible," he said.
Mancini rejected outright the notion that Sheikh Mansour would deliver a harsh judgment on two seasons of European failure.
"No. I don't fear this," said the Italian. "Why?
"We still have the Premier League and the FA Cup.
"If we think we can win the Champions League after two years we are crazy."
Mancini's reasoning is that City require more experience at this level before they can truly compete for the greatest prize.
"Chelsea tried to win Champions League for 10 years," he said.
"In the best moments, when Didier Drogba was young, they didn't win.
"They did it last year when no-one thought they could.
"We need to improve our team because there are a lot of teams better than us.
"I am not saying if you get to February you can't win. Anything can happen. But we didn't do that."
Madrid boss Mourinho was asked for his take on events at former club Chelsea on the day that Roberto Di Matteo was sacked and replaced by Rafael Benitez on an interim basis until the end of the season.
"That's football," Mourinho said. "I'm never happy when a manager is sacked. It can happen to me and is a feeling that no one likes to have.
"But at the same time, everyone knows I like Chelsea.
"I supported Roberto and I wished him to do well, and I will support the next chelsea manager."
"I like my old clubs to win," he said.
"I always wish the managers of my former clubs well so I always wish good luck to managers of my former clubs.
"I wish good luck to Benitez."