A native of Mexico, the righthanded closer was acquired at the 2018 trade deadline from Toronto for Ken Giles. Osuna was serving a suspension at the time for a domestic violence incident that took place in Canada while a member of the Blue Jays. The charges were eventually dropped but activists against domestic violence have continued to follow his exploits.

The nephew of former big leaguer Antonio Osuna, the 2017 All-Star went 2-2 with a 1.99 ERA and 12 saves down the stretch for Houston in 2018 but gave up a costly grand slam in the ALCS against Boston. The following year, he led the league with 38 saves and posted a 4-3 record with a 2.63 ERA but gave up four runs in ten innings pitched in the 2019 postseason, including a two-run homer in the ninth to D.J. LeMahieu in Game 6 of the ALCS that led to Jose Altuvé's dramatic walk-off clincher.

As an Astro, he is 6-5 with a 2.46 ERA and 50 saves in 89 games during the regular season. Through 2019, his career numbers are 14-18 with a 2.75 ERA and 154 saves over five seasons.