Glenn Murray's extra-time header, albeit from a marginally offside position, secured a famous 2-1 victory for the Eagles who put on a dogged display throughout the quarter-final tie.
Sir Alex was full of praise for Palace's effort, but hugely disappointed by his own side's performance.
"I don't know where to start to be honest. My apologies go to our fans tonight because that was not a Manchester United performance," he told MUTV.
"I don't want to take anything away from Crystal Palace - I wish them every success and I hope they get to the final. It was a fantastic, mammoth effort from all of them. Every one of their players worked their socks off to get to the semi-finals.
"We maybe played too many short passes, but I'm not going to dwell on it and go into the ins and outs of it all.
"When it went to extra-time I thought we must win it, but it wasn't to be. Their second goal was offside from what I've seen on the cameras, but I can't take it away from Crystal Palace, they fought hard to get the result and they deserve it. It's a very disappointing night for Manchester United."