"Whilst he was unable to guide us to safety, it certainly wasn't due to a lack of effort, professionalism and dedication," a Stoke statement said.
The Potters added that they plan to name a new boss "as soon as possible".
"The club would like to thank Paul for his work over the past four months and wish him well for the future," the club statement said.
"Paul would like to thank the board, players and staff for the opportunity and their support. He would also like to thank the fans for their unwavering support. Whilst looking forward to his next challenge, he would like to wish the club every success next season."