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.

In 2007, Osuna pitched in the Triple-A Mexican League for the Tigres de Quintana Roo recording a 1.61 ERA in 22 games and being named to the midseason All-Star team. He played for them again in 2008, but recorded a 7.94 ERA in only 11 games.

On August 30, 2011, his nephew Roberto Osuna was signed by the Toronto Blue Jays. Osuna was only 16 years old at the time of signing. He had been ranked fourth by Baseball America in projected signing bonus rankings.