The 34-year-old had more recently been playing his club football with Chinese side Shanghai Shenhua following his release from European champions Chelsea last summer.
However, having spent six months away from Europe's top competition, Drogba revealed that he cannot wait to play in the tournament once again.
"The opportunity to play for this great club was an offer that I could not turn down," said Drogba. "I am looking forward to playing in the Champions League again, against the best clubs in Europe."
Galatasaray will take on German side Schalke 04 in round two of the Champions League next month.