Roscic, 52, is currently president of Sony Music's classical music division and has never run an opera house. He was born in Serbia and grew up in Austria.
"I believe we need to look to the future, although that isn't to criticise the status quo," Culture Minister Thomas Drozda said as he announced the appointment.
He said he wanted to opera house "to ensure that it can attract the global stars of tomorrow... If you like, a State Opera 4.0."
In the age of Broadway and Netflix, "the genre has lost meaning and... faces never-before-seen competition for the public's time and money," Drozda said.
Roscic said accepting the job was "the most important decision of my life". He told a press conference he intends to "lift the quality of every night" so that the opera house will "create its own audience" and be able to compete with other media.