Cunniff was born in South Boston, Massachusetts and attended Boston College, where he was a two-time All-American left wing, and was the 1964 Rookie of the Year, and was the first hockey player ever to be named the Beanpot MVP twice ('64 & '65); a record that lasted for 39 years. He was a dominant scorer, his 71 goals and 82 assists in just 75 games established a Boston College career point record. His 67 points in 1966–67 was best in the nation, and helped the Eagles reach the NCAA Championship finals.