Oaks was traded to the Royals on Thursday as part of a three-team deal.
The White Sox received Oaks and minor-league infielder Erick Mejia while shipping Scott Alexander to the Dodgers and Joakim Soria to the White Sox. Oaks was bothered by an oblique injury last season but was able to compile a 3.64 ERA over 84 innings with Triple-A Oklahoma City. The 24-year-old righty will compete for a spot in the starting rotation with his new club.