The France international forward has signed a contract until June 2020 after Bayern agreed to pay the reported €21million fee required to buy him outright.
"Kingsley Coman is an important building block for the future of our team, so we decided to take up the option," chairman Karl-Heinz Rummenigge said via the club's official website.
"Kingsley is a player with great potential. We're convinced that he will help us in the coming years."
Coman joined Bayern on a two-year loan deal in 2015, with the Bundesliga champions possessing the option to finalise a permanent deal at any time before April 30.
The total cost of the transfer includes the two instalments which make up the €7m Bayern paid to Juve during the loan.
There had been reports in France that the 20-year-old was having second thoughts about the prospect of committing his long-term future to the Allianz Arena.
He has made 23 appearances in all competitions this season, scoring twice.