Drahman joined the Birmingham Barons in the double-A level Southern League for the 1990 campaign, spending the entire season with the club. He had a 6–4 record over 50 games (49 in relief), along with a 4.08 ERA and a 1.26 WHIP, striking out 72 in 90.1 innings pitched. In 1991, he opened the season with the White Sox. He blew a save in his first appearance then gave up five earned runs in 0.2 innings his next time out before settling down to earn a win on April 21. In that contest, he struck out one and allowed no hits or walks in the ninth before Lance Johnson hit a walk-off single in the bottom half for a 5–4 win over the Detroit Tigers. Overall, he played 28 games for the Sox, going 3–2 with a 3.23 ERA and a 1.109 WHIP, striking out 18 in 30.2 innings.
