Mancini's men led arch-rivals Manchester United 2-0 heading into the half-time break at Wembley on Sunday.
A second-half collapse saw the first piece of silverware for the 2011/12 season snatched from their grasp, with a stoppage-time winner from Nani handing United a 3-2 success.
A setback against the reigning Premier League champions has provided a less than ideal warm-up for the new campaign, but Hart believes City will learn valuable lessons from the loss.
"I think we needed a kick up the backside before the season started," said the England international goalkeeper. "We've done really well in pre-season but we have to learn that we're not quite there yet - but we will be.
"We never want to lose and we're not going to get used to losing. The feeling hurts but we're going to move on from there.
"We were 2-0 up against the run of play. I thought United played really well.
"I don't think we had the right mind-set when we came back out, I thought we defended a bit deep, but we have to try and take some positives away. We scored two good goals and we haven't really lost anything by losing this game."