Darren Ambrose proved Palace's inspiration, firing in a 35-metre wonder-strike after 65 minutes which television pundit and former Manchester United defender Gary Neville said was the best goal he had seen in 20 years at his old stomping ground.
Ambrose then turned provider when his wicked free-kick was nodded home by Glenn Murray, who appeared marginally offside, on 98 minutes to set the London team on their way to a clash with fellow Championship club Cardiff.
Premier League leaders Manchester City will face Liverpool in the other semi-final, to be played over two legs early next year, after they beat Arsenal and Chelsea on Tuesday.
"You can hear the fans out there now. We're delighted. We've got a great team spirit and we can beat anyone," former Newcastle United midfielder Ambrose told Sky Sports.
Match-winner Murray added: "It's what every boy's dream is, to come to a place like this and score a goal like that."
Palace's 1-0 advantage thanks to Ambrose lasted just three minutes when Italian Federico Macheda levelled from the penalty spot after he was hauled to the ground by captain Patrick McCarthy.
However United, fielding an inexperienced side in the often unloved competition, were lacklustre despite players supposedly pushing themselves for first-team inclusion and Palace deservedly inflicted a rare home defeat on Sir Alex Ferguson's men.
"I'm so disappointed. I have to apologise to my fans and all the players who weren't playing. We never expected that," said Ferguson, who saw Dimitar Berbatov, Rafael and Fabio come off injured.
"Any time we go on the pitch it means a lot to Manchester United. We have our pride to protect and our history to protect. Every time we walk on the pitch it's important. We didn't see that tonight."
