Darren Bent's 94th-minute equalizer saw United lose further ground in its bid for a Champions League place, leaving it nine points behind fourth-place Liverpool.
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"We should have won," Moyes told Sky Sports. "A couple of lapses cost us a bit. I think we had to defend probably three or four times for the whole game. Sometimes that happens in football. We did everything to try and win, we crossed it, we passed it, we just didn't have the finishing touches."
Moyes pointed the blame at his side's wastefulness in front of goal and felt that Fulham was lucky to get a point.
"I hardly felt as if they were up the park at all today," he added. "They didn't deserve to [get the early goal] and we had deserved to be in front. We didn't take the chances we had.
"If we hadn't got back in the game, there would have been something up. We deserved to win the game, we deserved to get the three points. Sometimes you don't get what you deserve."
Moyes said he thought his side "could have taken a bit more care with some of our crosses and deliveries" and declined to concede that United's bid to finish fourth was over.
When asked if Liverpool was too far away to catch, he simply offered: "We'll do our best."