Despite dominating possession and sending in a remarkable 81 crosses, Manchester United F.C. trailed for much of the match after an unmarked Steve Sidwell volleyed home Lewis Holtby's clever chip.
Two goals in two minutes from Robin van Persie and Michael Carrick appeared to have salvaged three points, but Darren Bent scored with a header in the dying minutes to grab a share of the points for Fulham.
"Today was as bad as it gets," Moyes, who replaced fellow Scot Alex Ferguson this season, told reporters.
"How we didn't win, I have no idea. It's goals that count. You can have as much possession as you like."
The defeat left Manchester United F.C. 15 points behind leaders Chelsea and in a battle to qualify for next season's Champions League.
They remained nine points adrift of fourth-placed Liverpool, and face missing out on Europe's premier club competition for the first time in 19 years.
The formidable aura that surrounded Manchester United F.C. under Ferguson has been eroded by losing eight league matches this season, half of those at home, and Carrick said defeat by a team who had lost their last four league matches was hard to stomach.
"It's a tough one to take, it's hard to explain how we've only come away with a point," he told the club's website (www.manutd.com).
"It was an incredible game, so one-sided. We played well for most of the game, we created opportunities but just couldn't score. When we finally got in front, we should have extended the lead with the chances we had, so it's tough to take after being hit like that at the end.
"We certainly had chances to do it (go further ahead), but even so, we were so in control that letting it slip like that is criminal."
Manchester United F.C.'s next game is on Wednesday at second-placed Arsenal who will be smarting from a 5-1 drubbing at Liverpool F.C. on Saturday.