Michael Carrick's deflected effort in the 80th minute appeared to have given United a much-needed victory but Fulham substitute Darren Bent was on hand to nod home after David de Gea had parried into his path deep into stoppage time.
After the match, United boss David Moyes felt his side deserved a win as luck again deserted the side who have won just once in their past four league outings.
"We should have won it but lapses cost us. We probably had to defend three or four times in the whole game, but that happens sometimes in football," he told Sky Sports.
"We tried everything to try and win - we crossed it, passed it, tried to get through - but we just didn't quite have the finishing touches.
"I think if we had got a goal early on then it would have completely altered the game. As it was, the more we went to score, they had a couple of chances and one of them they scored from. But I hardly felt as if they were on the park today.
"It has been quite often here at Old Trafford when we haven't got the goal, we have then maybe lost a goal and ended up chasing the game. We did it today.
"If we hadn't got back into the game then there would have been something up because we totally deserved it. We deserved to win the game, but you don't always get what you deserve."
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