The New Zealander, currently in charge of Super Rugby club the Chiefs in his homeland, will join the province in July on a three-year contract.
Rowntree's abrupt departure by mutual consent was announced on October 29.
McMillan coached the Maori All Blacks to a win and a loss against Andy Farrell's Ireland in 2022, before leading an All Blacks XV to victory over Munster at Thomond Park in November.
"I am humbled to be given the opportunity to lead Munster, an iconic club steeped in history," McMillan told Munster's website.
"Leading the All Blacks XV against Munster last year gave me insight into the immense pride and passion of the club's players, staff and wider rugby community. It was an incredible atmosphere and left a lasting impression.
"I am excited by the club's ambition and core values. They align strongly with my own, and I look forward to joining the team in July."
Head of rugby operations Ian Costello will remain interim head coach until the end of the season, before taking on the newly created position of general manager.
As part of the restructure, attack coach Mike Prendergast, who applied to succeed Rowntree, will move up to the role of senior coach during the summer.
Munster chief executive Ian Flanagan said: "After a thorough recruitment process, we are delighted to secure Clayton McMillan as our new head coach.
"Throughout our conversations with Clayton, what really stood out was his experience and how his coaching philosophy and values align with what we want to achieve at Munster Rugby."
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