Hangeland turns from villain to hero with Nani's aid
Simon Davies had earlier equalised a sweet strike from Paul Scholes, but things looked desperate for Fulham when, after Hangeland unluckily put the ball in his own net, United were generously awarded a penalty.
Goalkeeper David Stockdale saved Nani's weak spot-kick, though, and Hangeland headed home Damien Duff's corner to delight the Craven Cottage crowd.
"Straight after I scored that goal I looked at the clock and saw five minutes to do it," said Hangeland. "It was great to be able to score from that corner.
"It's a terrible feeling (to score an own goal). I don't think I could have done anything different. It looked like someone was going to head it.
"I had one chance to make up for it.
"We have had a great team spirit for the last two or three years. We never say die at this club and we showed it again."
Stockdale, in the team for former Middlesbrough goalkeeper Mark Schwarzer who continues to be linked with a move to
Arsenal, was Fulham's other hero for saving Nani's penalty after Duff was penalised for kneeing the ball against his own hand.
"It's a great feeling," said the goalkeeper. "It was guesswork. I just tried to get the lads out of the deep muck, and Brede came up trumps."
Asked if he could be Schwarzer's long-term replacement, Stockdale said: "I just take it day by day."
United manager Sir Alex Ferguson swallowed his disappintment to acknowledge Fulham were worthy of a share of the spoils.
"In fairness they deserved a point," he said. "They played very well in the second half.
"The first half belonged to us but they played better than us in the second half.
"When you have the lead and you have the opportunity to seal the game with a penalty you think you're home and dry, but I've got to give credit to Fulham."
Ferguson felt Stockdale's save from Nani's spot-kick was a key moment.
"The goalkeeper reacted quickly and you've got to give him credit, but I think it was a bad height," he said.
On Scholes's strike, he added: "It's never easy to score form 20-odd yards. He has a tremendous record and once again he's done it.
"Again, he was one of the best players on the pitch."