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He debuted in the 1964 Broadway production of Bajour, co-starring Chita Rivera and Nancy Dussault. Goz played Tevye in the Broadway musical Fiddler on the Roof from 1966 to 1968, both as understudy and lead actor. He also appeared in musicals such as Two by Two, and Chess and comedies such as Prisoner of Second Avenue. For his performance as Molokov in Chess he received warm reviews and 1988 Drama Desk Award nomination as an Outstanding Featured Actor in a Musical.
Goz had a number of TV and movie guest appearances over the years. He starred as The Big Apple in Fruit of the Loom underwear TV commercials during the 1970s. He starred in Bill, a 1981 television film. In his later years, Goz became known to a new audience with his role as the voice of Captain Hazel "Hank" Murphy in the Cartoon Network Adult Swim series Sealab 2021, a role he kept until his death from multiple myeloma at the age of 71 in 2003.