Green was born in the Bronx to Hungarian immigrants Daniel and Helen Weiss Green. Attending public schools, he wrote poetry, acted, and enjoyed reading, though not schoolwork. He attended New York University to study drama, meeting Comden in 1938.
Green married his third wife, actress Phyllis Newman, in 1960.
Green was survived by Newman.
His Broadway memorial, with such luminaries as Lauren Bacall, Kevin Kline, Joel Gray, Kristen Chenowith, Arthur Laurents, Peter Stone, and Betty Comden and Green's daughter Amanda, took place at the Schubert Theater December 4, 2002.