Clevenger exited in the top of the third inning after Andrew McCutchen's foul ball came directly back and hit Clevenger's bare hand.
Clevenger immediately crumpled over, cradling the hand, and he was attended to quickly by Seattle's athletic training staff. He walked off the field and was replaced by regular starter Chris Iannetta, who was getting the day off.
Clevenger was appearing in his 22nd game this season.
Seattle has just one catcher at Triple-A Tacoma, Mike Zunino, that is on the 40-man roster. The Mariners have said they would like to keep Zunino in Triple-A for the majority of this season to solve some of his hitting problems.
The Associated Press contributed to this report.