Three members of Patrick’s coaching staff - Osian Roberts, Kristian Wilson and Saïd Aïgoun - have also left the club.
Crystal Palace made the decision after a 12-game winless run.
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.
Chairman Steve Parish said: “It is with enormous regret that this difficult decision has been made. 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.
“That said, Patrick’s impact since joining us in the summer of 2021 has been significant, and he is held in the highest regard by myself, and all of his colleagues. He led the team to a Wembley FA Cup semi-final and respectable 12th placed finish last season playing some exciting football, which was a challenging and crucial campaign for the club given the changes we made to the squad prior to his arrival.
“Patrick has given his all to the club, and we all thank him and his team for their service.
“We also wish Patrick, Osian, Kristian and Saïd the very best for their futures.”
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