Cowen was born in Melbourne, and attended Scotch College before going on to the University of Melbourne. His studies were interrupted by World War II, during which he served in the Royal Australian Navy. After the war's end, Cowen attended New College, Oxford, on a Rhodes Scholarship. He subsequently won the prestigious Vinerian Scholarship as the best student in the Bachelor of Civil Law degree. He remained at Oxford after graduating, serving as a fellow of Oriel College from 1947 to 1950.