Greenwood played college football at the University of Missouri and the University of Illinois, where he starred as a halfback, punter and quarterback. His first year in professional football was with the NFL's Rams in 1945, when the team won the NFL championship. The Rams moved to Los Angeles after that year, however, and Greenwood elected to stay in Cleveland, where a new team called the Cleveland Browns was under formation in the AAFC. He played two seasons for the Browns, during both of which the team won the league championship. Greenwood was plagued by injuries in 1946, and in 1947 he suffered a debilitating cheekbone fracture that ended his professional career. After retiring, he worked as a high school coach in Ohio and briefly as an assistant at Yale University. In 1951, became the head football coach at Toledo University in Toledo, Ohio, but resigned from that position before completing his first season, saying the school had not done enough to counteract unnecessary violence in the game.