In 2012, he played for the Single-A West Virginia Power, logging a .280/.324/.454 batting line with career-highs in home runs (16) and RBI (72).

In 2015, Osuna split the season between Bradenton and the Double-A Altoona Curve, hitting a cumulative .286/.329/.435 with 12 home runs and 81 RBI in 129 games between the two teams.

He split the 2016 season between Altoona and the Triple-A Indianapolis Indians, posting a .279/.331/.457 slash line with 13 home runs and 69 RBI.

On November 26, 2020, Osuna announced on Instagram that he had signed with the Tokyo Yakult Swallows of Nippon Professional Baseball.

[citation needed] After the 2021 season, Osuna signed a three-year contract extension with the Swallows worth $5.1 million.