He is the son of the late Soviet-era and Russian actor Mikhail Kozakov and his father's first wife, Soviet-era costume designer Greta Taar. He, in turn, has given them two grandchildren: grandson Anton, from his first wife, Yuliya, and granddaughter Mariya, from his second wife, Russian actress Alyona Yakovleva.
He has one sister, Ekaterina (b. 1957), and, through his father, two half-sisters, Manana (b. 1969) and Zoya (b. 1996), and one half-brother, Mikhail (b. 1989).