After pitching one game for the Class A Appleton Foxes, Drabek was promoted to the AA Glens Falls White Sox and was 12–5 with a 2.24 ERA.

In 1985, Drabek returned to AA and spent the entire season at Albany-Colonie in the Eastern League and finished with a 13–7 record with a 2.99 ERA with 153 strikeouts in 192+2⁄3 innings.

After starting the 1986 season at AAA Columbus, Drabek made his Major League debut on May 30, coming in relief for starter Joe Niekro in a 6–3 loss to the Oakland Athletics.

Following the 1986 season, the Yankees traded Drabek with Logan Easley and Brian Fisher to the Pittsburgh Pirates for Rick Rhoden, Cecilio Guante and Pat Clements.

[8] Drabek enjoyed his best years with Pittsburgh, from 1987 to 1992, during which time he regularly pitched over 230 innings and consistently finished in the top 10 in the National League ERA race[citation needed].

He went 22–6 with a 2.76 ERA in 1990 en route to winning the National League Cy Young Award[9] and leading the Pirates to the postseason (where they lost in the NLCS to the Cincinnati Reds).

[11] Drabek returned to professional baseball in 2010, accepting a position in the Arizona Diamondbacks system as the pitching coach for the Yakima Bears in the short-season Class A Northwest League.