Karim Benzema gave Real the lead inside the opening 10 minutes when he ghosted in behind Maicon to sweep home from close range, before Cristiano Ronaldo's delicate chip was scrambled off the goal-line.
Sergio Aguero was then awarded a second-half penalty after he was adjudged to have been felled by Alvaro Arbeloa - who received his marching orders for a second bookable offence - but it proved to be too little, too late for Mancini's side who have still yet to win a game in the competition this season.
"We're very disappointed," Hart told Sky Sports at full-time.
"We have had a few opportunities in this group where we could have done a lot better.
"It's tough tonight, obviously. I don't think the referee has had a great game, I'm not one to blame things like that but the ball was out of play for 20 minutes. I'm very surprised that they did that, but we're out of the competition."
The England international was left to bemoan Jose Mourinho's time-wasting tactics, in spite of the officials reducing the visitors to 10-men in the second half.
But the lapse in concentration from both wing-backs that allowed Benzema to give the Spanish giants the lead in the opening exchanges left him more frustrated.
"Of course, all goals are [disappointing]," Hart added.
"A free man from seven yards out, it's sloppy. I don't know if it was a blessing, we're good enough to get back into games but we just didn't quite have enough time tonight.
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"But we move on to a big game on Sunday."
Mancini's side have slipped to defeat at the Santiago Bernabeu and Amsterdam ArenA, in spite of leading in both fixtures.
They could only muster three draws in front of their home support and Hart admitted City have surprised themselves by how poor their campaign has been.
He said: "All games have cost us. We've lost our two away games and not won at home, so I think you've got to look at both opportunities. We led in Amsterdam and in Madrid, so it's a bad campaign for us.
"It's incredibly frustrating because we're a very good team, I think we've surprised ourselves in a very, very bad way in this campaign.
"Unfortunately we weren't able to do the business when it mattered and that is Champions League football."
After the game, Citizens captain Vincent Kompany took to his Twitter account to express his hope for the club's future in Europe despite their disappointing form this season.
He Tweeted: "The opportunity to claim our place at the top of European football will come again. No sad faces, just the hunger to improve."