As a student at Elmira (N.Y.) Free Academy, Davis was a high-school All-American in football and basketball. Widely recruited to play running back in collegiate football, he chose to attend Syracuse University, in part because it was the school of his idol, Jim Brown (Brown, Jim). Davis wore Brown's number 44 at Syracuse, and in his sophomore year there he led the Orangemen to an undefeated season and a national championship. Syracuse clinched the national title with a 23–14 victory over the University of Texas in the 1960 Cotton Bowl. The game was highlighted by Davis's two touchdowns, which earned him Cotton Bowl Most Valuable Player honours. He was named an All-American in both his junior and senior seasons at Syracuse, and in 1961 he was awarded the Heisman Trophy as the most outstanding player in American college football—the first African American so honoured.

Davis was selected with the first overall pick of the 1962 National Football League draft by the Washington Redskins, who then traded him to the Cleveland Browns, whose owner Art Modell planned to pair Davis with Jim Brown in the team's backfield. Davis never played a game for the Browns, however, as he was diagnosed with leukemia before the College All-Star Game in July 1962. He underwent a variety of treatments in an attempt to return to football, but they were all unsuccessful; he died in a Cleveland hospital in 1963. Davis was posthumously inducted into the College Football Hall of Fame in 1979.