Palace have not won a match in 2023 and were beaten 1-0 by Brighton in the Premier League on Wednesday, their third defeat in a row.
They are 12th in the table but are only three points above 18th-placed Bournemouth, who have a game in hand.
"It is with enormous regret that this difficult decision has been made," said chairman Steve Parish.
"Ultimately, results in recent months have placed us in a precarious league position and we felt a change is necessary to give us the best chance of retaining Premier League status."