Honeycutt began his UT career in 1973. He worked his way into the starting rotation as a freshman. In his junior year, he made the all-SEC team, hitting .370 and making 12 starts as a pitcher. In his senior year, Honeycutt dominated on the mound, going 8-2 with a 2.88 ERA. He was named All-American and finished his four years at UT with an impressive .377 batting average, which still ranks him as one of the top five hitters in UT baseball history. He was twice named to the All-SEC team and the All-Academic team.

Honeycutt was drafted as both a pitcher and a first baseman by the Pittsburgh Pirates in June of 1976. In his first game as a professional, he was the starting pitcher, he batted clean-up and he hit a home run in his first at-bat.

Honeycutt spent very little time in the minor leagues, making his way to the Major Leagues in his second year. He enjoyed 21 years in the Majors, from his first appearance in 1977 at Yankee Stadium, to his last appearance in 1997. Along the way, his journey took him to the Seattle Mariners, Texas Rangers, Los Angeles Dodgers, Oakland Athletics, and St. Louis Cardinals.

During his Major League career, Honeycutt was a starting pitcher for the first ten years, making the All-Star team in 1980 and 1983. He had his best year as a starter in 1983, winning 16 games and winning the American League ERA title with 2.42. His next ten years were spent as a relief pitcher, where he became one of the toughest left-handed specialists in the game.

He pitched in three consecutive World Series with the Oakland Athletics in 1988, 1989, and 1990, winning a ring in 1989.