The 31-year-old, who left English team Queens Park Rangers by mutual consent last month having spent the second half of last season on loan at Qatar's Al Gharafa, has signed a one-year deal with an option for a further year.
"I have heard a lot of good things about Kuban, first of all from my great friend (Kuban midfielder) Charles Kabore," said Cisse whose previous clubs included Liverpool, Marseille and Lazio.
"I have spoken a lot with my friends who play in Russia including Samuel Eto'o and Florent Sinama-Pongolle. They gave good reviews about the country and the level of the championship," he told the club website.
Cisse, whose previous side QPR were relegated from the Premier League last season, said his experience in European club competitions, including winning the Champions League with Liverpool in 2005, could help his new team.
The striker, who has not featured for the national side since 2011, dreams of returning to the France squad.
"I love my country, I love the feeling I experience when I put on the tricolore shirt. I really want to put all my efforts, try my utmost, to return once again to the national set-up," he said.
"I understand well that in order to do that I need to have a very good season with my new club."
Kuban finished fifth in the Russian top flight last season to earn a place in the qualifying stages of the Europa League.