A graduate of Northwestern University, Graham played tailback in his college years. Upon joining the Cleveland Browns of the All-America Football Conference (AAFC) in 1946, he was switched to quarterback.
During the AAFC's four-year existence, the Browns won the championship each year. Graham threw for 10,085 yards and 86 touchdowns, while rushing for 11 more.
The Browns joined the National Football League in 1950, and won the league championship in their first NFL season. Behind Graham's 88 touchdowns in the NFL, the Browns won two more league titles, in 1954 and 1955. In the 1954 championship game against the Detroit Lions, Graham ran for three touchdowns and passed for three more.
Graham had an astounding 105-17-4 record with the Browns before he retired after the 1955 season. His 86.6 career pass rating is one of the best of all time. In 1965 he was inducted into the Pro Football Hall of Fame.
In 1959 he was hired as head football coach at the Coast Guard Academy, where his team went undefeated in 1963. Between 1966 and 1968 he coached the Washington Redskins; who went 17-22-3 during that time period. He then returned to be athletic director of the Coast Guard Academy.
He also played professional basketball in 1946 with the Rochester Royals.
Graham died of a heart aneurysm in Sarasota, Florida on December 17, 2003.