Early in his career Baumann was touted as “a Herb Score with control”. A reliable pitcher, he was effective as a starter, set-up man and occasional closer. His most productive season came in 1960 with the White Sox, when he had a 13-6 mark and led the American League pitchers with a 2.67 ERA.
In an 11-season career, Baumann posted a 45-38 record with a 3.90 ERA and 13 saves in 244 games pitched (78 as a starter).
In 1952 Baumann received a $90,000 signing bonus from Boston Red Sox owner Tom Yawkey, who nicknamed him "Beau".