The Toffees took a 24th minute lead through Nikica Jelavic, but the Reds responded as Sylvain Distin’s weak back pass was pounced on by Luis Suarez who poked home an equaliser before Andy Carroll headed home late on.
Moyes told ESPN: "It’s really disappointing not to get through. We started the game well and we got our goal.
"I’d actually thought we’d weathered most of it in the second-half as well but we can’t hide the fact we gave away two terrible goals.
"A mistake for the first one and we gave a free-kick away which allowed them to score from for the second.
"We were our own fault today because we gave them those opportunities. I thought we just about weathered in the second-half when we gave away the goal."
The Scot was frustrated to concede after sloppy play in the second half as he believed the Toffees had held off the worst of the Liverpool attacks.
He added: "I think with any goals you always look to try and correct them and do something about them. We’d done all the jobs we’d done well in the game, they missed one chance with Andy Carroll at the back post early on.
"With the rest of it we were coping with it fine. It happens and that’s why it happens if you make mistakes in games then it doesn’t win you trophies. Unfortunately today we made mistakes which probably gave us no chance of being in the final.
"Liverpool started brightly I thought we’d got through the worst of it we were just getting back and growing into it again. You’d hope we might knick the second-one. As it was we just got a wee bit of control and they get the goal."
Moyes felt Everton had done enough to force extra-time, however Carroll’s header four minutes from time condemned the Toffees to defeat.
The 48-year-old admitted he was not concerned with maintaining seventh position in the Premier League to stay ahead of Liverpool after Saturday’s defeat.
He said: "It looked for a lot of reasons it was going to extra-time but we gave away a free-kick which allowed them to score from it.
"We’re not thinking like that today. We’re thinking we’ve been knocked out of the semi-final of the FA Cup that’s the only thing we’re concerned with.
"Obviously we’re very sorry and disappointed for our supporters who came here after they way we’ve been we’ve been playing well but today we just succumbed to couple of poor decisions."