The London club will send Plange straight back to Derby until the end of this campaign.
He has signed a four-year contract that will keep him with the Selhurst Park club until 2025.
He said: "I'm very excited to be joining a big Premier League club. I'm ready to crack on.
Palace chairman Steve Parish said: "Luke is an exceptionally talented young footballer and one we hope will bring a range of attributes to the club in the years to come.
"But first, he has an important role to fulfil playing at Derby, and we appreciate his significance to them in their current circumstances. It is less than 12 years ago that our club was in a similar situation - as a result we were determined to ensure both clubs benefit through this deal. Therefore Luke will return on loan for the remainder of the season and help Derby with their important battles on and off the pitch.
"I look forward to seeing Luke compete in Palace colours soon, and until then enjoy the performances he continues to give for Derby."