Bevington was born in Akron, Ohio. His family moved to Santa Monica, California where he was a standout high school baseball player at Santa Monica High. He spent seven seasons in the minor leagues after being drafted by the New York Yankees in 1974. He batted .247 in 368 games played, including 33 with the Triple-A Vancouver Canadians of the Pacific Coast League in 1980. He threw and batted right-handed, stood 6 feet 2 inches (1.9 m) tall and weighed 190 pounds (86 kg).
He then managed in the Milwaukee Brewers organization for seven years before being named third base coach in 1989. In the middle of the 1995 season, Bevington's seventh on the ChiSox' staff, he was named manager of the White Sox when Gene Lamont was fired on June 2. After 2½ seasons at the helm, he was fired at the end of the 1997 with a record of 222-214 (.509); the White Sox went 75-76 in 1995, 85-77 in 1996, and 80-81 in 1997 under Bevington. He was a third base coach for the Toronto Blue Jays from 1999 through 2001 before returning to the minor leagues as a manager. He resigned as the Edmonton Cracker-Cats' skipper after a suspension stemming from an on-field brawl between the Cracker Cats and Calgary Vipers.