The 25-year-old midfielder passed his medical test in the Turkish city of Antalya, where the Russian club are currently preparing for the resumption of their league season, and signed a four-and-a-half-year contract with them.
"The club wishes to thank the international midfielder for his professionalism and his irreproachable attitude, which he has shown throughout the five seasons spent at Marseille," reads a statement from Marseille.
"I'll never forget the OM fans," Kabore told RMC radio. "I'll continue to support the team. It'll always be the club in my heart. I wish them all the best.
"I leave with a happy heart. One day, I'll be able to say to my grandchildren, 'I was lucky enough to have played at a big club like OM'."
Kabore was the captain of Burkino Faso at the 2013 Africa Cup of Nations.