Stevenson attended Vashon High School. He was the first African-American to receive a football scholarship from the University of Missouri.[1] As a sophomore, he registered 77 carries for 307 yards (third on the team) and 4 touchdowns. As a junior, he had 60 carries for 267 yards (third on the team) and one receiving touchdown.
He became a starter as a senior, as part of a backfield known as the "Fearless Foursome" that also included Mel West, Donnie Smith and Norm Beal. He posted 85 carries for 610 yards (second on the team), a 7.2-yard average and 6 touchdowns, contributing to an undefeated team (11-0 after a later forfeit by the University of Kansas) that won the Big Eight Conference title and the 1961 Orange Bowl, 21–14 over the United States Naval Academy. He also helped defeat the University of Oklahoma 41–19, rushing for 169 yards with touchdowns of 77 and 60 yards, which moved the Tigers to the top of the national polls for the first time in school history.
He finished his college career with 222 carries for 1,184 yards, 5 rushing touchdowns, 11 receptions for 168 yards and 2 touchdowns. During his time the team had a 22-9-1 record, including two trips to the Orange Bowl.