In what was his first FA Cup game in charge of the club, Moyes saw his men battle back from an early Wayne Routledge opener to go in level at the break.
However, United collapsed in the final 10 minutes with Fabio dismissed for a rash tackle of Jose Canas before Wilfried Bony grabbed a last-gasp winner to dump the Premier League champions out of the competition at the first hurdle.
Despite this disappointment, Moyes claimed United played well enough to at least earn a replay.
"We are a bit unlucky to lose but we probably didn't do enough to win it," Moyes told BT Sport.
"We made some opportunities, not enough, and when you're down to 10 men you want to see it out and take a replay but we didn't quite get that far.
"We played quite well at times, we got near the edge of the box quite regularly but didn't make the opportunity to score a goal or when we did get an opportunity take it. That was the disappointing thing, as we went behind and came back.
"We had certain periods of good control but we didn't get a goal or create enough chances.
"When we went down to 10 men, we kept trying to go for it and got caught a bit."
Moyes, who also saw Rio Ferdinand limp off in the match, acknowledged that losing five home games this season has not been good enough.
However, he remains confident the return of injured star performers such as Wayne Rooney and Robin van Persie will herald a change in fortune.
"The sooner we get them back, the better. But they’re not ready yet," the former Everton boss added.
"Our results at home haven’t been good enough."
Get the best features, fun and footballing frolics straight to your inbox every week.
Thank you for signing up to Four Four Two. You will receive a verification email shortly.
There was a problem. Please refresh the page and try again.