Derby will receive £2million in compensation for the 44-year-old, who replaces Paul Lambert at the Potters.
Rowett said: “This has come around quite quickly, but I am absolutely delighted to make the move to such an ambitious football club.
“This is a club that has had amazing stability over the years, has been established in the Premier League for some time and the challenge for me now is, of course, to get Stoke City back into that division as soon as possible.
“My ambition has always been to one day manage in the Premier League, and having been in the Championship for three years now I feel this is a great opportunity for me to do that.
“There are certain things to do now, and to get right, but the undoubted ambition of myself and the people above me is to take this club back up as soon as quickly as we possibly can.”
Rowett signed a new £1.5m-a-year Derby deal in January when Stoke made him their No1 target to replace the sacked Mark Hughes.
Rowett faced slashing the wage will at Pride Park next season with 22-goal striker Matej Vydra set to leave.
Now he faces the job of rebuilding Stoke as they look to bounce back from the Championship at the first attempt.
Stokes chiefs Peter and John Coates said: “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.”
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