Sanches was part of Portugal's Euro 2016-winning squad and joined Bayern from Benfica for 35million euros in May 2016.
But the 20-year-old has found game-time with the Bundesliga champions limited and made only 26 appearances last season.
Swansea boss Paul Clement spent time as Carlo Ancelotti's assistant at Bayern last season and used his contacts in Bavaria to pull off the shock move.
Clement has been seeking a creative influence since the departure of Gylfi Sigurdsson to Everton for a club-record £45m fee two weeks ago.
Sanches was named the best young player at Euro 2016 and joined Bayern after being linked with Manchester United among others.
Chelsea, Juventus, Liverpool and Monaco were also linked with Sanches this summer amid speculation over his future.
But Bayern refused to sanction a permanent transfer and were only interested in allowing him to go out on loan.
"We wanted Renato Sanches to go to a club where he could play at a high level," Bayern chairman Karl-Heinz Rummenigge told German sport magazine Kicker.
"Renato has to play continuously for a year, which is not guaranteed at the moment in the squad."
Sending Sanches to Swansea also appealed to Bayern as they are not a Champions League rival.
Sanches has completed the necessary paperwork while on international duty with Portugal, and will join the Swansea squad before their next Premier League game against Newcastle on September 10.
Clement hailed Sanches as "one of Europe's elite young talents" as he welcomed him to his squad.
"I'm delighted to secure his services,'' Clement told the official Swansea website.
"He is a player I know well having worked with him for six months at Bayern Munich before my own move to Swansea.
"He is one of Europe's elite young talents who has already achieved a lot for his age, including winning Euro 2016 with Portugal.
"Having moved from Benfica to Bayern Munich, he has not played as much as he would have liked. But this move to Swansea and the Premier League will provide him with a real positive development experience.
"As a player he is a dynamic, powerful central midfielder who I believe has all the attributes to suit the Premier League.''
Related links
Fast Results
Scores
