The Red Devils crashed out of the competition on penalties at Old Trafford, after winning 2-1 on the night to make it 3-3 on aggregate.
"We did not play well enough, we had the chances to put it to bed but didn't. Sunderland played well and they deserve credit but overall we looked like we would get there until the last minute," BBC Sport quotes Moyes as saying.
"I was disappointed, but we did not play well enough in the end. They did not give us too many problems, we might have scored on the break but we couldn't, and it cost us. That's football, we'll get on with it, pick ourselves up and go again."
Sunderland will now face Manchester City in the League Cup final.