Kapustin was born in the town of Gorlovka in eastern Ukraine on 22nd November, 1937. At the age of 14 he moved to Moscow and began piano lessons with Avrelian Rubakh and later Alexander Goldenweiser.

Kapustin’s first attempts at composition began with a piano sonata at the age of 13. During his studies at the Moscow Conservatory Kapustin composed and premiered his first opus, a Concertino for piano and orchestra (1957). During this time he also had his own quintet and was a member of Yuri Saulsky’s Big Band. After graduating in 1961 Kapustin became a member of the Oleg Lundstrem Big Band.

From the early 1980s he focused completely on composing. Kapustin uses jazz idioms in formal classical structures.

His music is often performed by leading pianists including Marc-André Hamelin, Steven Osborne, Ludmil Angelov, Masahiro Kawakami, Nikola Petrov and Vadim Rudenko, and by cellists including Eckard Runge and Enrico Dindo.