Ryan Mmaee's own goal condemned the Potters to a fourth successive loss in the second tier, against a team that went into that match bottom of the table.
The result also extended Stoke's winless run to six matches, a streak which stretches back to October.
That form has left Stoke 20th in the Championship table, just two points clear of the relegation zone, and the club have now taken the decision to sack Neil following 16 months at the helm.
"Alex is a man of absolute integrity who has given his all for Stoke City, and we would like to thank him for his hard work during his time with the club," said chairman John Coates in a statement.
"We are grateful for the building blocks he has helped put in place in bringing together a group of players in whom we have a huge amount of faith regarding their ability to turn things around this season, and who can help us achieve longer-term success.
"However, with the way the results have been so far, we have made the difficult decision to seek a new direction for the team at this time.
"Nothing matters to me more than the success of our club and we are now working towards the appointment of Alex's successor."
Paul Gallagher will take the reins as caretaker boss, assisted by Alex Morris, Ryan Shawcross and Jonathan Gould.
Technical director Ricky Martin added: "The recruitment process is now underway and we will explore all avenues to secure the best person for the job.
"The role represents a fantastic opportunity and I anticipate that it will attract a number of high-calibre candidates.
"With Martin Canning having also left the club with our sincere thanks for his hard work as Alex's assistant, Paul Gallagher has been installed as caretaker manager ahead of the forthcoming home fixture against Swansea City.
"Paul will be assisted by Alex Morris, Ryan Shawcross and Jonathan Gould. We wish them all well for Tuesday evening's game."
Alex Neil took over the reins at Stoke in August 2022 and oversaw 66 matches as manager, winning 22 of those and losing 31.
Stoke are next in action at home to Swansea City on Tuesday night.