Drogba had been tipped to make the move to the Far East since he announced last month he was leaving Chelsea after scoring the decisive penalty for the London club in their Champions League final shootout victory over Bayern Munich.
The 34-year-old would join up with his new team-mates in July once his contract with Chelsea expired on June 30, Shanghai said in a statement on Wednesday.
"I have studied all the offers I had during the past few weeks and I have the feeling that going to Shanghai is the right direction for me now," Drogba said on his website.
"I am looking forward to that challenge, to discover a new culture and I am very enthusiastic about the development of the Chinese football league."
Shanghai will be hoping the Ivorian, nicknamed 'The Beast' in China, can make an instant impact and help the side arrest a dismal start to the campaign which has left them sitting 12th in the 16-team league despite heavy investment.
Ambitious Shanghai owner Zhu Jun posted a photo on his microblog site of Drogba and the club director Zhou Jun posing with a Shenhua shirt, with the caption "Our warmest welcome," shortly after the powerful forward announced the move.
"The twin guns of Chelsea will now don the jerseys of Shanghai Shenhua to do battle at Hongkou Stadium," Shanghai said on their website.
"This scene would perhaps only be played out on a fantasy football game. But today, this dream has become a reality. With hard work and determination, a dream will always become reality one day."
Drogba's arrival is the latest in a long list of high-profile names who have made the move to China in recent months. Italian World Cup winning coach Marcello Lippi leads a Guangzhou Evergrande side packed full of skilful South American talent.
But Drogba, who scored 157 times in 341 appearances for Chelsea making him the club's fourth all-time leading scorer, is perhaps the biggest catch.
He spearheaded Chelsea's Champions League titl
