Former Cardiff player Joe Ledley scored Palace's second goal, in between strikes from Jason Puncheon, as the Londoners continued their spectacular revival under manager Tony Pulis.
"It was a big game for us, and I am very disappointed for the fans and players that we ended up with a performance like this," Cardiff manager Solskjaer said.
"If you produce a miracle - that sometimes happens in football - you are still capable of getting out of this and staying in the Barclays Premier League.
"I am full of fight. It has been a tough season for everyone involved with the club. Today in the second half, it showed a little bit that it has taken its toll.
"There was a lack of belief a little bit towards the end. We ran out of ideas.
"Players are desperate to perform. I have seen them all week, and you are so disappointed for them and the fans that we produced a performance like we did. Of course they want to stay in the league."
For breaking stories and all the great banter like us on Facebook: facebook.com/tribalfootball