Drogba has signed an 18-month contract in Turkey after quitting Shanghai Shenhua.
Drogba, 34, said: "The opportunity to play for this great club was an offer that I could not turn down.
"I am looking forward to playing in the Champions League again, against the best clubs in Europe."
Galatasaray chairman Unal Aysal added: "The negotiations had been going on for a couple of days.
"This morning we did the last negotiation and this evening we have done the deal.
"I talked to Drogba on the phone and he is so glad to join a big team like Galatasaray."