Nicknamed "El Cañón" (The Cannon) in his native Mexico, Osuna signed with the Los Angeles Dodgers in 1991, and made his major league debut with them in 1995, appearing in 39 games for them that year. Going into 1995, he had been the Dodgers #2 prospect and #15 overall as rated by Baseball America. Osuna posted earned run averages of 3.00, 2.19, and 3.06 over the next three seasons. On March 17, 2001, he was traded to the Chicago White Sox with minor leaguer Carlos Ortega for Gary Majewski and minor leaguers Andre Simpson and Orlando Rodriguez. Osuna Spent time on the disabled list in 2001 and played only 4 games for the White Sox. In the 2002 season, Osuna appeared in 59 games with a 3.86 ERA. On January 15, 2003, he was traded to the New York Yankees with minor leaguer Delvis Lantigua for Orlando Hernández.