Rangnick has signed a two-year deal and will take up the role alongside his consultancy job at Old Trafford.
The Austrian FA (OFB) held an executive committee meeting on Friday morning, with the 63-year-old's appointment now given formal approval by board members.
At the start of April, the OFB officially denied approaching Rangnick while it was reported Austria sporting director Peter Schottel had travelled to Manchester to offer him the job.
The national team were on the lookout for a new boss after Franco Foda resigned following Austria's 2-0 defeat to Wales in their World Cup qualifying play-off earlier this month, which denied them a spot at the World Cup in Qatar.
Now, with Erik ten Hag confirmed as the next permanent Manchester United manager, Rangnick has been cleared to take on the job with Austria.
And his deal with the European nation will be extended automatically by a further two years should they qualify for the 2024 European Championship.
"I will take over as national team manager of Austria at the end of the season but will continue my consultancy with Manchester United," Rangnick said.
"I'm really looking forward to playing my part in helping United become a real force again.
"It is an honour for me to take on the role of [Austria] manager," he added.
"The prospect of contesting the European Championships in Germany with a young team hungry for success fills me with great anticipation."
Gerhard Milletich, President of the Austrian Football Association, said: "We are very pleased that in Ralf Rangnick we have been able to recruit an outstanding expert in international football as team manager.
"We are all convinced that he is the ideal man and that his vision will advance the national team and the OFB."
Rangnick replaced Ole Gunnar Solskjaer at Manchester United at the end of November, agreeing to take temporary charge of the team until the end of the campaign.
The Red Devils are currently sixth in the Premier League, five points behind Arsenal in the fourth, with Champions League football next season now looking unlikely.