Tcherepnin studied at the St. Petersburg and Tbilisi Conservatories, and among his teachers were composers Nikolay Alexandrovich Sokolov and pianist Leocadia Kashperova. After his family moved to France in 1921, Alexander finished his studies in Paris with composer Paul Antonin Vidal and pianist Isidore Philipp, head of the piano department at the Paris Conservatoire.
Alexander worked closely with the Parisian composers school, namely Arthur Honegger, Bohuslav Martinů, Marcel Mihalovici and Conrad Beck. He began yearly visits to the USA in 1926 and made several extended trips to China and Japan between 1934 and 1937. Tcherepnin worked with Akira Ifukube, Yoritsune Matsudaira, Fumio Hayasaka, Kō Bunya, and He Luting, and established a publishing house in Tokyo to promote their music. While in China, he met the Chinese pianist Lee Hsien Ming (1915–1991), and the two later married in Europe. They had three sons together: Peter, Serge and Ivan.
In 1948, they relocated to the United States and settled in Chicago, where Alexander taught composition at DePaul University (1949–64). His students there included Phillip Ramey, Robert Muczynski, Gloria Coates, and John Downey. The Chicago Symphony Orchestra premiered his second symphony with Rafael Kubelik conducting. In 1967, Alexander toured in USSR and gave concerts in Moscow, Leningrad and Tbilisi.