Unis was a standout golfer at Parma earning four varsity letters. In 1964, he won the Ohio State Amateur Championship and became a PGA Tour professional in 1965. In 1967, he qualified to compete in the US Open.
Throughout his storied professional career, he won many more professional championships and honors. He competed in the PGA senior tour and won the PGA Senior Player of the Year Award in 1990. He also won the Wisconsin Senior PGA Championship twice. Unis has served as a teaching professional at numerous courses over his long career. He is also a member of the Parma Amateur Athletic Federation Hall of Fame.