Postecoglou signed a three-year contract with his home-town club, who finished a disappointing eighth in the Australian league in the recently concluded season.
"This is a great opportunity and presents a whole new challenge and one that I'm really excited about, especially considering I'm a Melburnian, and I can't wait to get started," Postecoglou told reporters on Thursday.
Postecoglou had a hugely successful two-and-a-half year stint with Brisbane, who he led to a 36-game unbeaten run, the longest by a team in any Australian sport, as well as becoming the first side to win back-to-back titles.
"Ange has been the outstanding coach in the Hyundai A-League over the past two seasons," Melbourne chairman Anthony Di Pietro told reporters.
"He implemented a very progressive style of play at Brisbane Roar and we believe he is the right person to take our club to the next phase of success."
Postecoglou's assistant at the Roar, Rado Vidosic, was named as coach of the Brisbane side on Wednesday.
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