Mane leaves Bavaria after just one season during which he made 38 appearances across all competitions and scored 12 goals.
Mane won the Bundesliga and the German Super Cup in his short time at the Allianz Arena but his stay was not deemed a successful one.
The 31-year-old missed the 2022 FIFA World Cup in Qatar through injury and landed a club fine in April following an altercation with teammate Leroy Sane.
Al Nassr have paid £34m to sign Mane for the next three years and the former Liverpool forward will join the likes of Cristiano Ronaldo and Marcelo Brozovic at his new club.
“We want to thank Sadio Mane for the past season,” Bayern chief executive Jan-Christian Dreesen said after the Bundesliga champions announced his departure.
“It certainly wasn’t an easy year for him, getting injured just before the World Cup and being unable to take part in it with Senegal, who he’d previously led to a first triumph at the Africa Cup of Nations and World Cup qualification.
“Because of his long lay-off, he also couldn’t have the impact at FC Bayern that we all and he himself had hoped for. That’s why we came to the joint decision that he’ll begin a new chapter in his career and make a new start at a different club. We wish him all the best and lots of success for the challenges ahead at Al-Nassr.”