Mane has signed a four-year contract, Al-Nassr added on messaging platform X, formerly known as Twitter.
Financial details were not disclosed but media reports said the Saudis paid 40 million euros ($43.85 million) for the 31-year-old who had two more years left on his Bayern contract.
Mane moved to Bayern last year with high hopes after netting 120 goals in 269 games for Liverpool where he won the Champions League, Premier League, FA Cup, League Cup and Club World Cup.
He was a key figure in the Merseyside club's attack but failed to make the same impression at Bayern, where he fell out with team mate Leroy Sane and hit the German winger - an offense for which he was fined and suspended.
The twice African Footballer of the Year averaged more than 20 goals a season in his last five campaigns with Liverpool but scored only 12 for Bayern.
"We want to thank Sadio Mane for the past season," Bayern CEO Jan-Christian Dreesen told the club website.
"It certainly hasn't been an easy year for him, suffering an injury just before the World Cup and missing out on participation with Senegal, whom he had earlier led to their maiden Africa Cup of Nations triumph and World Cup qualification.
"Due to the long downtime, he wasn't able to contribute as much to FC Bayern as we and he himself had hoped. That's why we decided together that he should start a new chapter in his career and make a fresh start at another club."
Mane joins an Al Nassr outfit who have strengthened their squad since the marquee signing of Cristiano Ronaldo last term.
The Saudis, who were runners-up last season, have signed midfielders Marcelo Brozovic from Inter Milan and Seko Fofana from Lens while left back Alex Telles has also joined from Manchester United.
Al-Nassr were banned from registering new players earlier this month due to outstanding debts, with FIFA saying they had to pay 460,000 euros ($504,712.00) related to the transfer of Nigerian Ahmed Musa from Leicester City.
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