The 20-year-old midfielder has been with the Eagles since the summer, since then he has made eighteen appearances for the club and has more than impressed the powers that be at Selhurst Park.
He joins Dougie Freedman’s side on a two and a half year deal.
Marrow told the Palace website: "I have enjoyed my time here and have been impressed with the squad's quality.
"I am looking forward to working hard and improving my game."
Dougie Freedman said: "I've been impressed by Alex since I saw him in a reserve game for Blackburn and he has really shown his quality during his games in a Palace shirt.
"It's good to see him join the club permanently and I am sure he will become an important part of our squad for some time to come."