The 44-year-old has put pen to paper on a three-year contract, officially becoming Paul Lambert's replacement at the bet365 Stadium.
Club owners Peter and John Coates said in an official statement: "We are delighted to be able to move so quickly to secure Gary's services and we are now looking forward to working with him towards our objective of returning to the Premier League as soon as possible."
Rowett will move straight to the former Premier League club from Derby County, where he spent 14 months of his career and guided the team to the Championship playoff semi-finals, where they lost to Fulham.
The Rams released a statement saying: "Rowett confirmed that he wished to speak to Stoke about the possibility of joining them and both clubs have subsequently agreed compensation terms in order for him to move to the bet365 Stadium.
"We wish to place on record our thanks to Gary for his contribution during his time as manager. The process for recruiting a new Manager is underway and the club will make further comment as and when it is appropriate to do so."
Stoke picked up just seven Premier League wins all season.