Galatasaray announced on their website on Monday that Drogba had agreed to an 18-month deal, cutting short his stint in China where he had been playing for Shanghai Shenhua since leaving Chelsea after last season.
The deal is worth 10 million euros ($A12.87 million) the club announced, plus a bonus of E15,000 ($A19,300) for each match played.
The announcement comes shortly after Sneijder completed his move from Inter Milan, as the Turkish club has now made two biggest signings of the January transfer window.
Drogba is currently with Ivory Coast at the Africa Cup of Nations in South Africa, but said on Twitter through his PR company: "I am looking forward to playing in the Champions League again, against the best clubs in Europe."
Drogba played a key part in Chelsea's run to the Champions League title last season, scoring a late equaliser in the final and then converting the decisive penalty in a shootout against Bayern Munich.
"The opportunity to play for this great club was an offer that I could not turn down," Drogba said.
His representatives, the Sports PR Company, said he will join up with the Galatasaray squad after the African Cup is over.
Galatasaray faces German club Schalke in the round of 16 of the Champions League, with the first leg to be played in Turkey on Feb. 20.
Drogba joined the Shanghai club in July but his lucrative deal soon became part of a boardroom dispute at the club, and the striker had previously asked for a loan deal during the Chinese off-season.