He was nicknamed "The Mad Hatter" because of his habit of frantically tugging at the bill of his cap and re-adjusting it prior to every pitch as well as "Third Degree" for his penchant of giving up runs near the end of his career.

Burns made his MLB debut on May 31, 1988, and during that season was a reliably effective starter at the back end of the A's rotation.

He compiled an 8–2 record over 14 starts and three relief appearances as the A's won the AL pennant but then lost the World Series to the Los Angeles Dodgers.

He helped the A's win two more pennants in 1989 and 1990, and a World Series in 1989, when the A's swept their cross-bay rivals, the San Francisco Giants.

Burns posted a regular season won-loss record of 21–23 with a 3.47 earned run average and 13 saves in 203 games pitched.