His representatives revealed on Twitter that the Ivory Coast international would join up with his new side following the African Nations Cup and Gala confirmed the deal on their official website.
Drogba memorably converted the winning penalty in the final shootout against Bayern Munich to win the Champions League for Chelsea for the first time last season. It turned out to be his last kick for the London club.
Explaining the reasons behind his move to Turkey, Drogba said through his representatives on Twitter: "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."
An emotional reunion with Chelsea is not possible, though, after the holders crashed out of the competition in the group stage.
Drogba's capture is the second marquee signing within a week for Gala, coming hot on the heels of the arrival last week of Holland midfielder Wesley Sneijder from Inter Milan.
Drogba's move to Turkey brings to an end a six-month stint in China.
The 34-year-old joined Shanghai last June, penning a two-and-a-half year deal reportedly worth £200,000 a week, but was repeatedly linked with a move back to Europe in recent months.
Last month Drogba returned to train with Chelsea during the Chinese off-season in preparation for the African Nations Cup.
FIFA rejected a special request from the striker for him to join a club on loan outside the January transfer window.
Despite his protests to the contrary, reports persisted that Drogba's move to China had not worked out and he has now left the Far East to return to Europe.
Gala are five points clear at the top of the Super Lig standings and are through to the last 16 of the Champions League, where they face Schalke.