Choupo-Moting, 31, was a free agent at the end of last season after his contract expired at Paris Saint-Germain.
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The striker has signed a contract until June 2021 with the Bundesliga champions.
"It's a great feeling coming back to the Bundesliga -- and then to the biggest club in Germany," he told the club's website.
"Who wouldn't want to play for FC Bayern? It's an honour to play for this club. It's always Bayern's aspiration to win everything, and I'm hugely motivated to help them achieve those goals."
Choupo-Moting became an unexpected hero for PSG after scoring the winning goal in their Champions League quarterfinal tie against Atalanta in August.
The striker also came off the bench in the Champions League final as Bayern overcame PSG to win their sixth European trophy.
The winger played for Bayern between 2015-17 where he won two league titles before moving to Turin.
"I'm very happy to be playing for FC Bayern again," Costa said. "I had a wonderful time in Munich with lots of successes and I'm sure we'll win titles again."
The German champions also completed the signing of defender Bouna Sarr from Marseille on a four-year deal, while loaning Mickael Cuisance to the French side for the one season. Marc Roca also joined Bayern from Espanyol on Friday.