Sutton attended UT on a baseball scholarship and played for the Texas Longhorns, where he was a two-year letterman under coach Cliff Gustafson. His teammates at UT included future Major League pitchers Roger Clemens and Calvin Schiraldi. Sutton spent three years as a backup second baseman, with just 50 at-bats in his career prior to the 1983 playoffs. On a hunch, Gustafson put Sutton in the post-season line-up as a left fielder; Sutton hit .454 in the six-game tournament and was named regional MVP as the Longhorns went on to win the College World Series. The 1983 team finished with a 66-14 record.

In various interviews, Gustafson has named Sutton his all-time favorite player. "He sparked us to a regional win and continued to spark us through the national championship," Gustafson said in 1994. "Clemens and Schiraldi got all the hype, but Sutton was the key to the national championship run."