Glenn Murray headed an extra-time winner as the Championship outfit claimed a memorable 2-1 quarter-final win at Old Trafford.
Ferguson, who had made 10 changes for the tie, paid tribute to the visitors but was far from impressed by the performance of his own players.
He told MUTV: "I don't know where to start to be honest. My apologies are to the fans because that was not a Manchester United performance.
"I don't want to take anything away from Crystal Palace. I wish them every success and hope they get to the final because it was a fantastic, mammoth effort from them tonight. Every one of their players worked their socks off to try to get to the semi-final. I am just very disappointed."
Palace took the lead with a stunning goal from Darren Ambrose on 65 minutes but United looked the most likely to progress after Federico Macheda replied from the penalty spot.
Yet Palace dug deep and grabbed the winner in the 98th minute when Murray, despite being close to straying offside, nodded in an Ambrose free-kick.
Ferguson said: "When it went to extra time I thought we must win but it was not to be.
"Obviously the goal was offside - the second goal - from what I have seen.
"We are not getting any luck from linesmen just now but I can't take that away from Crystal Palace. They fought hard to get the result and they deserve it."
Goalscorer Ambrose savoured a "great occasion".
"It's fantastic," Ambrose told Sky Sports 1.
"It was a great occasion coming here against the champions.
"It was a massive occasion for the fans."
Cardiff await in the last four after the draw was made following the match, meaning one npower Championship club will take on either Manchester City or Liverpool at Wembley.
Wednesday night's win was made all the more remarkable by a poor run of Championship form, which has seen Palace go five games without a win or a goal.
"We've had a quiet period in the season but we came here in confident mood and the spirit is fantastic," Ambrose continued.
"We've got a great team spirit, it's a lot different from last year - we can beat anyone, and we showed that tonight [Wednesday].
"We weren't worried about the lack of goals, we've got goalscorers and that showed tonight [Wednesday], it was a great finish from Glenn on the second."
Manager Dougie Freedman, who has earned plaudits for his work since taking over a cash-strapped club midway through last season, added: "It's a wonderful night for the club, especially given the position we've been in the last few years.
"To play at Old Trafford tonight [Wednesday], we're privileged, the lads learnt a lot from tonight [Wednesday].
"I think we deserved it. We played some stuff, some good football.
"We thought if we came with a gameplan that could be stuck to and everybody played eight, nine, 10 out of 10 we could do it."