Cooper honed his skills at Cedar Crest and turned professional in 1923. His first pro win, the Galveston Open in 1923, came before he turned twenty years of age. A perennial U.S. Open contender (with seven top-10 finishes and second place in 1927 and 1936), Cooper was nicknamed "Lighthorse Harry" by sportswriter Damon Runyon for his quick pace of play in winning the inaugural Los Angeles Open in 1926, completing the final 18 holes in 2.5 hours. (During the American Revolution, Lighthorse Harry Lee was a cavalry officer in the Continental Army, later the father of Confederate general Robert E. Lee.)