Lambert, who failed to save the side from relegation to the Championship, left Stoke by mutual consent on Friday.
They had been heavily linked to Rowett, who was previously in charge of Derby County. The Englishman led Derby to the Championship play-offs, where they lost to Fulham, and he had been reported to have asked permission to talk to Stoke over the vacant manager's position.
The 44-year-old was confirmed as the new manager on Tuesday, as the 'Potters' announced that he had signed a three-year contract.
“We are delighted to be able to move so quickly to secure Gary’s services," Stoke chief Peter Coates said. "We are now looking forward to working with him towards our objective of returning to the Premier League as soon as possible.”
Rowett made it clear that the main objective was to return Stoke to the Premier League: "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."
"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 quickly as we possibly can.
"We have to make sure that the guys who report back for the first day of pre-season first and foremost want to be at Stoke City, and that they are willing to put in the desire and effort that not only I want, but that Stoke City and the supporters of this club want as well."
May 22, 2018