The former Chelsea striker, who had asked to leave the Chinese club after they failed to pay his reported £200,000 per week salary, has signed an 18-month contract with the Istanbul giants.
Galatasaray's statement to the stock exchange specified Drogba would earn 6 million euros for his season and a half at the club, a signing-on fee of 4m euros and 15,000 euros per match played.
The club added that the Turkish Football Federation had requested an international transfer certificate from the Chinese FA.
"The opportunity to play for this great club was an offer that I could not turn down," Drogba said.
"I am looking forward to playing in the Champions League again, against the best clubs in Europe," added the Ivory Coast captain, who scored the decisive penalty when Chelsea won last season's final.
LigTV, the official broadcaster of the Turkish league, had prematurely announced the deal just before the Galatasaray-Besiktas derby, but it appears an agreement in principle had already been struck.
It is the second massive deal for the Turkish giants this January with Wesley Sneijder having already signed from Inter Milan.
Drogba had been playing in China with Shanghai but his future there was plunged into doubt amid reports of unpaid wages.
He signed a two-and-a-half-year deal after leaving Chelsea last June, but the Shanghai-based Oriental Sports Daily said the 34-year-old was owed wages in the wake of a long-running equity row at the club.
Drogba applied for permission in November to leave Shanghai on loan before the January transfer window, but FIFA refused the request.
He then returned to Chelsea to train at his former club in a bid to stay sharp for the African Cup of Nations, where he is currently in action for the Ivory Coast.