Drogba has signed a two-and-a-half year deal reportedly worth £200,000 a week with the Chinese Super League side. He will join up with former Chelsea colleague Nicolas Anelka, who joined the club in January.
"I am looking forward to a new challenge, to experiencing a new culture, and excited about the new developments in the Chinese Super League. When Chelsea went to China last year, we had a great time and I met some amazing fans.
"I hope to help promote Chinese football around the world and further improve the links between China and Africa."
Shanghai owner Zhu Jun said: "We have been wanting to sign Didier for a while, and are very happy that it has finally happened. He is an incredible player, who has won so many trophies in his career, we feel that his experience and ability will be a very valuable asset to our squad.
"Shanghai is one of the most influential cities in the world, and we are very excited to welcome Didier here."