After winning the semi-final second leg at Old Trafford after extra time and levelling the tie at 3-3, United missed four of their five spot-kicks in the penalty shootout to crash out of the tournament.
It appeared that Sunderland would advance in extra time after Phil Bardsley's 119th minute goal but Javier Hernandez equalised seconds later to force the tie into penalties.
"If we were going to get through, we were going to stumble over the line, not because we played well enough," Moyes said.
"We didn't play well enough to merit going through - Sunderland came here and deservedly got themselves in the final."
Sunderland keeper Vito Mannone saved Rafael's penalty to send his side into the final with Danny Welbeck, Adnan Januzaj and Phil Jones also fluffing their spot-kicks.
Moyes had no qualms with choice of penalty takers - with the players choosing the five takers themselves- but labelled the execution in the shootout as 'terrible' with several key players unable to participate.
"In the end, I just don't think we got the level of performance we needed to get through the game," he said.
"It went to penalties - and our penalties were terrible.
"Ideally, it wouldn't have been the group we had chosen - Javier (Hernandez) would have taken a penalty but was injured and people like Michael Carrick would have been involved.
"There's always going to be a loser in a semi-final and it happens to be us because we didn't execute our penalties well enough."
In better news for United, it appears that Chelsea playmaker Juan Mata is close to joining the club for a reported fee of £37million.
When asked about Mata during the press conference, Moyes replied: "We are not saying anything."
