United enjoyed the lion's share of possession but could not turn their dominance of the ball into the solid currency of three points and were undone in the 94th minute when Fulham's Darren Bent grabbed a late leveller.
"We deserved to get back in the game and we deserved to win it, but we gave away a diabolical second goal," Moyes told reporters.
"We should have made it 3-1 and it was really disappointing that we didn't do that. We should have seen the game out.
"I don't know if we could have done an awful lot more. We completely dominated the game and should have won it comfortably."
He added on MUTV: "We needed to take the opportunities that were given to us, like some of the half chances.
"I don't think anyone can really say we did a lot wrong, we tried to play it, get it wide at the right times. What we needed to do was get the third goal when we were 2-1 up."
The draw was a blow to United's hopes of qualifying for next season's Champions League and left them in seventh place, nine points adrift of Liverpool in fourth.
United's leaky defence were undone after 19 minutes when Steve Sidwell was left unmarked to sidefoot home a pass from Lewis Holtby.
The champions set up camp in the Fulham half but did not find a breakthrough until the 78th minute when Robin van Persie finished from close range.
When Michael Carrick's deflected effort gave them the lead it looked like United were set to complete a deserved victory, but out of nowhere Bent headed home in stoppage time to leave Moyes scratching his head for answers once again.
"I was probably the most relieved person (when we went 2-1 up) because I thought 'my goodness' as we needed to win the game," Moyes said.
"Even in the last five minutes Fulham weren't even coming up for the ball, we were keeping the ball and they weren't necessarily trying to score an equaliser - but somehow they did...
"But maybe there were a few things today that we didn't quite do well enough. Maybe our finishing wasn't quite good enough and bits of our defending weren't quite good enough either."
(Reporting by Toby Davis; editing by Rex Gowar)
