McGraw was born in Garden City, Kansas. Standing 6'5" and weighing 235 lbs., McGraw was fresh from action with the U.S. Marine Corps in World War II when he enrolled at Colorado State University in 1946, when it was known as Colorado A&M. Working diligently at his game, McGraw called upon lessons learned as a boxer and wrestler to fashion himself into a special breed of football player. His arm strength was crushing to opposing players and his agility developed through wrestling served him well when fending off enemy linemen. Colorado A&M finished 2–7 in McGraw's freshman season, but things would change quickly. As a sophomore, McGraw helped his Rams post a 5–4–1 mark, the team's best record in 11 years. Then, in 1948, the Aggies posted upsets over rivals Utah State, Wyoming, BYU and arch rival Colorado. The 1949 campaign, McGraw's last as a four-year letterman, saw the Aggies log a 9–1 record marred only by a loss to Wyoming. After graduation, McGraw joined the Detroit Lions and captured Rookie of the Year and All-Pro honors. In 1981 he was inducted into the College Football Hall of Fame.
While at Colorado State, McGraw was the school's first consensus football All-American in 1948 and 1949. When he returned to CSU in later years as a staff member, he also served as the athletic director from 1976 to 1986.