At age twenty, Alexanian settled in Paris, where he met Pablo Casals. Casals had seen Alexanian perform, and noticed that Alexanian's fingering was in line with his new way of playing the cello. They got to know one another, finding they had similar views on general technique and interpretation of music.
In 1921, Alexanian became the assistant to Casals at the École Normale de Musique, which Casals had founded with Alfred Cortot, August Mangeot, and Jacques Thibaud in 1919. There, he and Casals put their revolutionary ideas into practice. Students from around the world came to study with him at this time, including Gabriel Cusson, Maurice Eisenberg, Antonio Janigro, Gregor Piatigorsky, Hidayat Inayat Khan, Pierre Fournier, and Emmanuel Feuermann. During his tenure at the school, Alexanian published his 1922 book on cello technique, Traite Theorique et Pratique du Violoncelle, as well as his famous edition of the Bach Suites in 1929.
Alexanian abandoned his position at the École in 1937, and moved to the United States. There he taught both at the Peabody Institute in Baltimore, and the Manhattan School of Music in New York City. There his students included Bernard Greenhouse, David Soyer, George Ricci, Raya Garbousova, David Wells, and Mischa Schneider.