Cosell was born on March 25, 1918 in Winston-Salem, North Carolina. He studied at New York University and at New York University School of Law. Cosell was married to Mary Edith Abrams from 1944 until her death in 1990. Cosell and Abrams had two children. Cosell died on April 23, 1995 in New York City, New York from heart failure, aged 77.