Kluber (4-4) stymied the Rangers on six hits, working around three walks while striking out three. The right-hander threw 69-of-104 pitches for strikes in notching his first win since May 15.
Mark Reynolds went 2-for-4 with a run-scoring single for the Indians, who posted only their second win in June and evened their three-game series.
Derek Holland (5-3) absorbed the loss, surrendering four runs on nine hits over 4 1/3 innings.
A.J. Pierzynski socked a solo homer for Texas, which came into the contest with a one-game lead over Oakland atop the American League West.