The Turkish club confirmed on their website that Drogba had agreed an 18-month deal.
The 34-year-old is currently on international duty for Ivory Coast at the Africa Cup of Nations.
Drogba - via his representatives, Sports PR Company, on Twitter - said: "I am looking forward to playing in the Champions League again."
Galatasaray are through to the last 16 of the competition, where they face German side FC Schalke 04.
