Cowen was born in Melbourne in 1919 into a Jewish family named Cohen, the son of Bernard Cohen (1892 – 1975), from Belarus and Sara Cohen (née Granat; 1893 –1989), born in Australia. His paternal grandparents were Jewish immigrants from Belarus in the Russian Empire. The name was formally changed to Cowen when he was a young boy. He was educated at St Kilda Park State School, Scotch College and the University of Melbourne. He served in the Royal Australian Navy during the Second World War, and was in Darwin during the bombing raid of 1942, before being transferred to Brisbane to work in the cipher unit of General MacArthur's office. He married Anna Wittner in 1945. Cowen then went as a Rhodes Scholar to New College, Oxford, where he completed a Bachelor of Civil Law degree and jointly won the Vinerian Scholarship. From 1947 to 1950 he was a fellow of Oriel College, Oxford, and was also a consultant on legal matters to the British Military Government in Allied-occupied Germany.