The head coach led the London side in 200 games across six seasons and two spells, but has seen form dip in recent months amid growing speculation of his departure at the club.
He was also taken ill last week on Friday but has now been released from hospital and is said to be doing well.
Hodgson said in a statement: “This club is very special and means so much to me and has played a big part in my footballing life. I have fully enjoyed my time here across six seasons, as it has given me the chance to work with top class players and staff doing what I love every day.
“However, I understand, given recent circumstances, it may be prudent at this time for the club to plan ahead, and therefore I have taken the decision to step aside so that the club can bring forward their plans for a new manager, as intended for this summer."
Chairman Steve Parish added: “Roy has a special place in Crystal Palace history and this will never be forgotten. After four years in which he led the club to maintaining Premier League status season after season, he once again joined us nearly a year ago to steady the ship, and worked wonders.
"That he then agreed to continue in the summer speaks volumes about his commitment to our club. Quite simply, we owe our continued Premier League status to Roy.
“I would like to thank Roy enormously for his service and wish him the very best for the future; it’s fair to say Roy has the keys to Selhurst Park and will always be welcomed back."
Paddy McCarthy (40) and Ray Lewington (67) will lead Palace against Everton - a game you can follow with Flashscore.
Former Eintracht Frankfurt boss Oliver Glasner (49) seems likely to take over at Palace with his name being linked heavily with the position even before news of Hodgson's departure was announced.