The Pittsburgh Pirates selected Cliburn in the fourth round of the 1977 MLB Draft out of Delta State University.

He spent five seasons in their Minor League system as a starting pitcher, before being released at the end of spring training in 1982.

He made his major league debut as a September call-up that year, appearing in one game against the Kansas City Royals, pitching two innings and giving up three runs.

After that, he had a brief stint with the Daytona Beach Explorers of the Senior Professional Baseball Association in 1990, where he formed a battery with his brother Stan.

Stew returned to their double-A affiliate New Britain Rock Cats in 2009 to aid the development of younger pitchers.