The deal is for 18 months starting in the second half of the current season, Galatasaray Sportif, the Istanbul's club merchandising unit, said.
"The opportunity to play for this great club was an offer that I could not turn down," Drogba was quoted as saying on Twitter.
"I am looking forward to playing in the Champions League again, against the best clubs in Europe," added the Ivory Coast captain, who won Europe's leading club competition with Chelsea last year when he scored the decisive penalty in a shootout.
The Galatasaray deal includes transfer fees totalling six million euros and payments of 15,000 euros per match, plus signing fees of 4 million euros.
The 34-year-old Drogba, on Africa Cup of Nations duty in South Africa with his country, moved to China after Chelsea's Champions League triumph last year.
Drogba applied for permission to leave on loan before the January transfer window but world football's ruling body FIFA refused the request.
The striker returned to Chelsea to train at his former club in a bid to stay sharp for the Nations Cup, fuelling media speculation of a return to Europe.
Drogba's transfer comes less than a week after Netherlands midfielder and 2010 Champions League winner Wesley Sneijder joined Galatasaray from Inter Milan.
Galatasaray have been drawn to meet German club Schalke 04 in the last 16 of the Champions League next month.
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