Moyes believes several decisions wrongly went against his side, most notably when Adam Johnson was awarded a penalty just after the hour mark after going down under pressure from midfielder Tom Cleverley, with Black Cats striker Fabio Borini making no mistake.
Referee Andre Marriner's decision to award the spot-kick drew the ire of Moyes, whose side slipped to their third successive defeat for the first time since 2001.
A clearly upset Moyes told Sky Sports News: "We're having to play them as well as the opposition at the moment. It's really terrible, it really is.
"We're actually beginning to laugh at them, that's the thing."
Defender Nemanja Vidic had drawn United level in the 52nd minute after Ryan Giggs scored an own goal to give Sunderland a half-time lead.
While Borini ensured Sunderland have a narrow advantage heading into the second leg at Old Trafford, Moyes was unequivocal when asked whether he felt any pressure to reach the final.
"None whatsoever," he said.
"This is a two-legged game and I'm actually really looking forward to this second leg now."
United had several chances to pull level once more, with the mercurial teenager Adnan Januzaj offering a lot of hope for the visitors, and Moyes felt his side were unlucky to come away from Wearside with a defeat.
"The players deserved it because they did well enough to get a result," he said.
"We'll do everything we can to try and get it right. The players put in a strong performance tonight, it just didn't quite come off.
"But it's not far away and it's coming, I can sense it.
"You just need to look at the performance of Adnan Januzaj tonight; he was terrific.
"He put in a great job and played really well."
Moyes could not call upon strike duo Wayne Rooney or Robin van Persie due to injury and the Scot is hopeful they will be available in the return leg.
"Hopefully we will get these players back because they'll be big players for us coming back. If we can get them back for the second leg that would be great," he added.