Gleaton was the Rangers first-round draft pick in June of 1979, signed out of the University of Texas. After the 1980 season, he was included in an 11-player trade to Seattle that featured Rick Honeycutt and Richie Zisk. Gleaton was released by the White Sox before finding a home as a set-up man in the 1987 Royals bullpen.
Gleaton stayed in Kansas City for three years, the last of which (1989) was marred by two demotions to Triple-A Omaha. Dealt to Detroit the following April, Gleaton found a niche with the Tigers and set a career-high with 13 saves.
Gleaton pitched well again the next season, but Detroit didn’t offer him enough money to stay in 1992. Although he remained confident in his abilities — the 34-year-old veteran told Baseball Weekly in March that he was throwing harder than he had been a decade earlier — a brief stint with the Pirates that season proved to be his last taste of major-league action.
