Kurt Yaghjian is the son of Haig Yaghjian, a former assistant conductor of the Cincinnati Symphony Orchestra, and was a member of a school choir that participated in the premiere of Gian Carlo Menotti's The Death of the Bishop of Brindisi in May 1963. He is of Armenian descent. Menotti was impressed with the expressiveness of Yaghjian's face and recommended him to NBC for the role of Amahl in Amahl and the Night Visitors in 1963, the first time that the production was videotaped, previous versions having been shown live. The videotaped version with Yaghjian was shown annually until 1966. In 1971 he graduated from North Carolina School of the Arts.
From the early 1990s until after the events of September 11, 2001 Yaghjian was in a band called Little Isidore and the Inquisitors. In October 2001 he joined another band called Kenny Vance and the Planotones. His musical appearances include Amahl in the television movie Amahl and the Night Visitors (1963), Annas in Jesus Christ Superstar (1973) (film and soundtrack), and Hair (1979).