Anzhi confirmed the deal on their official Twitter page (@fc_anji), saying: "Samba. All signed in time to enter the system. All data is docked."
The transfer window in Russia did not close until Friday and the Russian side, who recently appointed Guus Hiddink as their coach, have taken advantage to add Samba to their growing list of stars.
The 27-year-old made a transfer request in January but remained at Ewood Park after the winter window closed when a possible move to QPR or Tottenham failed to materialise.
Anzhi have made a number of notable signings of late, funded by billionaire owner Suleyman Kerimov.
Their squad already boasts Cameroon striker Samuel Eto'o, Russia international Yuri Zhirkov and former Brazil defender Roberto Carlos, while they have also been linked with a number of other high-profile players.
The France-born centre-back joined Rovers in 2007 from Hertha Berlin, and has made 161 appearances for the Premier League outfit, scoring 16 goals.He is also a Congolese international, with five caps to his name.
Samba will wear squad number 22 for the Premier Liga side.