McBride, who has held the role for six years, will step down and be replaced by Crozier on 29 November.
Crozier, 54, has previously held directorships at Saatchi & Saatchi, Vue International, G4S, the Football Association, the Royal Mail and most recently ITV, where he served as CEO until June 2017.
He also joined the Whitbread board in April 2017, becoming senior independent director in September 2017 and chair in February 2018.
Senior independent non-executive director, Ian Dyson, said: “On behalf of the board, I’d like to thank Brian for his contribution during a period of high growth for our company and I’d like to welcome Adam, who joins us at an equally exciting stage in our journey.”
McBride said: “It has been a great honour and pleasure to have worked with Asos over these past six years and I wish the company continued success in being the world’s number one destination for fashion-conscious twenty-somethings.”
Crozier added: “The Asos story is truly impressive and I am tremendously excited to become part of it. I’m greatly looking forward to working with the team that is building a global powerhouse in online fashion.”
Adam Crozier has no interest in any ordinary shares or options in Asos.