He has previously played in Major League Baseball (MLB) for the Toronto Blue Jays and Houston Astros.

He made his major league debut in 2015 at 20 years old, becoming the youngest pitcher in Blue Jays history and the first player born in 1995 to play in MLB.

[3] Osuna made his professional baseball debut with the Diablos Rojos del México of the Mexican League in 2011.

On May 9, 2013, it was announced that Osuna had torn his ulnar collateral ligament, an injury that usually requires Tommy John surgery.

[11] Despite his attempt to rehab his elbow, Osuna underwent Tommy John surgery in late June.

[13][14] Osuna made his first rehab appearance since the procedure on July 8, 2014, and pitched one inning for the Gulf Coast League Blue Jays.

[15] He was then promoted to the High-A Dunedin Blue Jays and made seven starts before the end of the season, posting a 0–2 record, a 6.55 ERA, and 30 strikeouts in 22 innings pitched.

[17] While initially not expecting to have any chance at making the major league team out of camp, Osuna impressed Blue Jays management through the first half of the spring.

[18][19] After Steve Delabar was optioned to minor league camp on March 26, it was reported that Osuna would likely make the Opening Day roster as a reliever.

[21][22] Osuna made his Major League debut in a game against the New York Yankees on April 8, 2015, striking out Alex Rodriguez and getting Stephen Drew to flyout.

[23] In making his debut, Osuna became the youngest pitcher to appear in a Major League game for Toronto, at 20 years and 60 days old.

[1] Through the first month of the season, Osuna became the most reliable arm in the bullpen for the Blue Jays, posting a 1.38 ERA through his first 10 appearances.

[26] On August 7, Osuna became the youngest player in MLB history to record an extra-innings save, when he closed out a 2–1 win over the New York Yankees in the 10th inning.

[30] During the offseason, the Blue Jays acquired reliever Drew Storen, who had been the Washington Nationals' closer for part of the 2015 season.

He also earned his first career postseason win, pitching two perfect innings in the Blue Jays' decisive Game 3 victory of the ALDS[28] On December 5, 2016, Osuna committed to play for Team Mexico at the 2017 World Baseball Classic.

[36] Osuna was diagnosed with a cervical spasm late in spring training and started the 2017 season on the 10-day disabled list.

[38] Osuna made his return to the mound the following day, pitching the final inning of the Blue Jays' 8–2 victory over the Kansas City Royals.

[44] On June 22, Osuna was suspended for 75 games, retroactive to May 8, due to violating the league's policy on domestic assault.

[45] On July 30, 2018, the Blue Jays traded Osuna to the Houston Astros for pitchers Ken Giles, Héctor Pérez, and David Paulino.

Due to Roman Cota's refusal to testify, the prosecution withdrew the charge against Osuna in exchange for a peace bond, mandating that for one year he not have contact with the alleged victim without her consent.

[48] [65]According to Toronto police sources, Roman Cota expressed her intention to resume contact with Osuna.