Clevinger is 0-1 with a 7.71 ERA in four appearances -- three starts -- for the Indians this season.
The AL Central leaders were outscored 35-16 in losing the first three games of the series against the last-place Twins. Cleveland's starting pitchers allowed 22 runs -- 21 earned -- in 8 1/3 innings.
Manager Terry Francona used 14 relievers for 18 2/3 innings in the first three games, and the overworked bullpen has led to a flurry of roster moves.
Left-hander Ryan Merritt was called up from Columbus while right-handers Cody Anderson and Shawn Armstrong were optioned to the Clippers before Thursday's game. Armstrong was recalled Wednesday and pitched two innings in a 13-5 loss.