United were frustrated by a Fulham side who sat back and allowed them to attack consistently, particularly after Steve Sidwell's early goal.
Last season's Premier League champions completely dominated the match but did not equalise until the 78th minute through Robin van Persie.
Michael Carrick scored what looked to be the winner less than two minutes later but Darren Bent ensured Fulham would claim a point with an injury-time header.
When asked if he was disappointed with his players' mental fragility, Moyes said: "Yeah – you could use maybe, maybe a sort of mental softness that we didn't see the job out and get the job done.
"So I would agree with that."
Moyes said his team were hurt by their latest slip-up, a result which leaves them nine points behind fourth-placed Liverpool.
"The players are hurting. I can see them every day, they hurt because the results aren't going the way they want," he added.
"They're really good professionals in the way they go about their work. The way I see them training makes me feel they will get results, there's no doubt about that."
United now face a massive test to catch Liverpool and finish fourth – and secure a place in next season's UEFA Champions League – but Moyes remains hopeful.
"We have to win all our games," he continued.
"So you can ask me (about Liverpool) as many times as you like. I'm going to keep saying we have to win the games.
"We should have won that one today. And we needed to as well, I felt.
"We've got a good team and there'll be very few teams desperate to play Manchester United, that's for sure."
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