[10] The Baltimore Orioles selected Cleavinger in the third round of the 2015 MLB draft, and he made his professional debut with the Aberdeen IronBirds, posting a 6–1 record with a 2.19 ERA in 19 games.

[11] Cleavinger spent 2016 with the Delmarva Shorebirds and the Frederick Keys, going a combined 7–3 with a 3.07 ERA in a career-high 76+1⁄3 innings pitched.

On July 28, 2017, the Orioles traded Cleavinger and Hyun-soo Kim to the Philadelphia Phillies for Jeremy Hellickson.

Cleavinger made his MLB debut on September 17, against the New York Mets, at Citizens Bank Park.

[20] On August 1, 2022, Cleavinger was traded to the Tampa Bay Rays in exchange for minor leaguer German Tapia.

[21] He logged 13 appearances for Tampa Bay to close out the year, posting a 2.41 ERA with 25 strikeouts in 18.2 innings pitched.

In a May 7 appearance against the New York Yankees, Cleavinger suffered a knee injury after tagging Aaron Hicks out in a rundown.