He debuted in the 1964 Broadway production of Bajour. Goz played Tevye in the Broadway musical Fiddler on the Roof from 1966 to 1967, both as understudy and lead actor. He also appeared in musicals such as Two by Two, Prisoner of Second Avenue, and Chess.
Goz had a number of TV and movie guest appearances over the years. He played The Big Apple in Fruit of the Loom underwear TV commercials. 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.