The Ivorian striker, 34, released a statement on his website, confirming he has signed a two-and-a-half year deal with the Chinese Super League club, and expects to join the team in July.
Details of the contract were not disclosed but it has been reported that he will earn around £200,000 a week in China.
"I have considered all the offers I have received in the past few weeks, but feel that Shanghai Shenhua FC is the right move for me at this time," Drogba said. "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."
Drogba had announced his decision to leave Chelsea after eight years at the club. In his final match for the Blues he scored the equaliser against Bayern Munich in the Champions League final, before slotting the winning penalty in the shootout.
He scored 157 goals in all for Chelsea after joining from Marseille in 2004.
He will link up with former Chelsea team-mate Nicolas Anelka at Shanghai Shenhua, with the club currently labouring in mid-table in the Chinese Super League.
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."
Shanghai's new manager Sergio Batista said: "Drogba is a very good soccer player, his outstanding skills and amazing experience will give us motivation to improve our games. It is my honour to invite him to join in our team. At last I have to say that the club owner Zhu Jun did his utmost to bring Drogba to Shenhua."