He was born in Saint Petersburg, Russia, and played the piano and composed prolifically from a very early age. He was stimulated in this activity by the atmosphere at home, which -- thanks to his family's Benois - Diaghilev connection -- was a meeting place for many well known musicians and artists of the day. By the time he began formal theory and composition studies in his late teens, he had already composed hundreds of pieces, including more than a dozen piano sonatas. Among his teachers in Russia were composer Victor Belyayev (pupil of Anatoly Lyadov and Alexander Glazunov), who prepared Tcherepnin for St. Petersburg Conservatory; Leocadia Kashperova (renowned pianist, protégée of Anton Rubinstein); and his professor at the Conservatory Nikolay Sokolov (pupil of Nikolai Rimsky - Korsakov). Notably at that time Tcherepnin’s mentor was famous musicologist Alexander Ossovsky, who also was a friend of his father. His works were influenced by composer Alexander Spendiarov.

After the 1917 Russian Revolution, the family fled St. Petersburg and settled for some time in Tbilisi, Georgia. In young Tcherepnin's luggage were some two hundred short piano pieces, quite a number of which eventually reached print (notably in his Bagatelles, Op. 5). In Tbilisi he continued his studies at the conservatory, gave concerts as both pianist and conductor and wrote music for the Kamerny Theater. Because of the increasingly hostile political environment in Tbilisi after Georgia was sovietized, the Tcherepnins chose to leave Russia permanently in 1921. They settled in Paris, where Alexander completed his studies with Vidal and Isidor Philipp, who was the head of the piano department at the Paris Conservatory, and became associated with a group of composers that included Bohuslav Martinů, Marcel Mihalovici and Conrad Beck. Philipp secured the publication of several groups of short piano pieces that Tcherepnin had composed in Russia. From Paris Tcherepnin launched an international career as a pianist and composer. In 1925 he won the Schott Prize with his Concerto da Camera, Op. 33. He began yearly visits to the United States in 1926 and later went to the Far East, making several extended visits to China and Japan between 1934 and 1937. He promoted composers in Japan (Akira Ifukube, Bunya Koh, and others) and China (He Lüting and others), even founding his own publishing house in Tokyo for the purpose. While in China, he met the young Chinese pianist Lee Hsien Ming, and the two later married in Europe. They had three sons together: Peter, Serge and Ivan.

During World War II, he lived in France. The war virtually stopped his musical activities. The immediate postwar period, however, brought a resurgence of creative energies; the result was a number of important works, beginning with Symphony No. 2 (composed 1947, not orchestrated until 1951). In 1948, he went to the United States, settling in Chicago in 1950 and in 1958 acquiring United States citizenship. He and his wife taught at DePaul University in Chicago, where the Chicago Symphony Orchestra premiered his second symphony with Rafael Kubelík conducting. His students there included Phillip Ramey, Robert Muczynski, Gloria Coates, and John Downey. In 1957, he completed two major American orchestral commissions: the Divertimento, Op. 90 (for Fritz Reiner and the Chicago Symphony Orchestra) and his Symphony No. 4, Op. 91 (for Charles Munch and the Boston Symphony Orchestra). In 1964 he moved to New York and subsequently divided his time between the United States and Europe. He died in Paris in 1977.

The Singapore Symphony Orchestra has recorded his first ever complete symphony cycle, conducted by Lan Shui. In 2008, these recordings were reissued together with Singapore Symphony performances of his six piano concertos (Noriko Ogawa, pianist), along with the Symphonic Prayer, Op. 93, Magna Mater, Op. 41 and other orchestral works.

He was a National Patron of Delta Omicron, an international professional music fraternity.

His early works were fairly original and some of his pieces have enduring popularity. His output includes three operas, four symphonies, a divertimento (which is a symphony in all but name), six piano concertos, works for ballet, choral music, alto saxophone solo, and a large amount of solo piano music. His Symphony No. 1 (1927) is remarkable for including the first symphonic movement ever written completely for unpitched percussion. One of two symphonies left incomplete at his death would have been for percussion alone. Tcherepnin invented his own harmonic languages. The most famous of his synthetic scales, derived by combining minor and major hexachords, has nine notes and consists of three conjunct semitone - tone - semitone tetrachords. This came to be known as the "Tcherepnin scale", and may be classified with Messiaen's modes of limited transposition.