Bessent was a pitcher for the Generals in the late 1940s. After graduation Bessent was signed by the New York Yankees where he played minor league ball for two seasons. After taking a year off in 1952 Bessent was drafted by the Brooklyn Dodgers. Bessent was finally called up by the Dodgers to the Major League in 1955, where he pitched over 3 scoreless innings in route to the Dodgers winning the 1955 World Series. Bessent played for the Dodgers until September of 1958 compiling a 14-7 record, 4.05 ERA and 118 strikeouts, mostly coming out of the bullpen. Bessent retired in 1962 from baseball.