Alexander Plisetski was born on 20 October 1931 into the family of a diplomat and an actress, both of Lithuanian Jewish origin.
His father, Mikhail Plisetski (1899–1938), was Consul General of the USSR at the island of Spitsbergen, where he managed the coal concessions (trust "Arctic - carbon").
During her imprisonment, Alexander stayed with the family of her brother, Asaf Messerer, while his elder sister, Maya Plisetskaya, who later became a famous ballerina, was placed in the custody of her sister Sulamith Messerer.
Plisetski was in need of heart surgery, which should have been carried out in the United States by invitation from Igor Youskevitch; however, due to his choreographic commitments, the operation was postponed.
In the last year of his life he was actively involved in the staging of Serenade for Strings, which premiered in Moscow in November 1984.