Born in Poland to a Jewish family, Drabowsky emigrated to America in 1938. He excelled as a pitcher in high school and college and was signed as a bonus baby by the Chicago Cubs. He debuted for the Cubs in 1956 and finished tied for second in the National League in strikeouts in his rookie season. In 1958, he gave up Stan Musial's 3,000th hit. An arm injury that year curtailed his effectiveness, and after a couple more seasons with the team, he was traded to the Milwaukee Braves. He played for the Braves, the Cincinnati Reds, and the Kansas City Athletics in 1961 and 1962 before remaining with the Athletics through the end of the 1965 season. During this period, he was sent to the minor leagues a few times, and while in the major leagues, he typically went back and forth between the starting rotation and the bullpen, except in 1963, the year he had his lowest earned run average (ERA) as a starter. Drabowsky also was the losing pitcher to Early Wynn in Wynn's 300th win that season. Following the 1965 season, he was selected in the Rule 5 Draft by the Baltimore Orioles.