The Reds defended stoutly for most of the game at Old Trafford on Wednesday night but an inability to add to Jonny Evans’ first-half goal meant the tie was still precariously poised going into extra time.
Ex-Red Phil Bardsley turned the game on its head with a late goal in the added period, only for Javier Hernandez to score a dramatic, 120th-minute tie-leveller.
But the Black Cats then prevailed 2-1 on penalties to book a Wembley date with Manchester City, leaving Moyes with no complaints.
The manager told MUTV: "We probably haven't played well enough tonight to merit going through, but we were getting there. We had hung on.
"We had defended well enough in the end to make sure we were going to see the game out at 1-0. But it changed, and credit to the lads. They got themselves a goal again.
"But I've got no complaints. I just don't think we had a level of performance that merited winning the game more comfortably, and that's what I was looking for.
"If we'd gone through I'd have been disappointed in the performance, but obviously I'm doubly disappointed because we