Bernard Zaslav was born in Brooklyn, New York and graduated from Julliard School of Music. He studied violin there with Sascha Jacobsen and Mischa Mischakoff. He later studied viola with Lillian Fuchs at Yale Summer School. In 1948 Zaslav joined the Cleveland Orchestra for two seasons and afterwards played in many prominent symphony orchestras in the New York area. His main area of expertise however, was chamber music. He played with the Kohon String Quartet in New York, the Fine Arts Quartet in Wisconsin, and the Vermeer Quartet in Northern Illinois where he taught. Additionally he joined the Stanford String Quartet at Stanford University in 1985 where he was on the faculty. In 1962 he created the Zaslav Duo (piano and viola) with his wife, Naomi Zaslav which has performed and recorded around the world.

Zaslav was also a firm advocate for new music and was one of the original members of the Composers Quartet which performed and recorded many new works of American composers at the time.