Cleavinger picked up the save in Saturday's extra-inning win over the Dodgers, surrendering one unearned run over one inning while not allowing any hits or walks.
Cleavinger (7-2) struck out one without allowing a baserunner in a scoreless eighth inning to earn the win Sunday against the Reds.
Cleavinger threw a scoreless 10th inning with one strikeout against the Yankees on Saturday to earn his first career save.
Cleavinger (knee) will not have any restrictions during spring training, Marc Topkin of the Tampa Bay Times reports.
Cleavinger, who underwent surgery May 17 to repair a torn ACL and meniscus in his right knee, is expected to be fully healthy in advance of spring training, MLB.com reports.
Cleavinger was transferred to the 60-day injured list by the Rays on Friday, Marc Topkin of the Tampa Bay Times reports.
Cleavinger is likely done for the season after being diagnosed with a torn right ACL, Marc Topkin of the Tampa Bay Times reports.
Cleavinger was placed on the 15-day injured list by the Rays on Monday with a right knee sprain, Marc Topkin of the Tampa Bay Times reports.
Cleavinger was recalled from Triple-A Durham on Saturday, Neil Solondz of the Rays Radio Network reports.
The Rays acquired Cleavinger from the Dodgers on Monday in exchange for minor-league outfielder German Tapia, Marc Topkin of the Tampa Bay Times reports.