The Indians ended an otherwise frustrating night by handing Texas closer Joe Nathan only his second blown save of the season.
Three of the four runs Cleveland gave up were unearned, including both of the Rangers' scores in the eighth that gave them a 4-2 lead.
Nathan (2-4), who had saved 33 games in 34 chances coming into this contest, was sent to the mound in the ninth and he gave up a leadoff homer to Carrera.
Pinch-hitter Russ Canzler then singled off Nathan and Kipnis followed by hitting a 1-0 pitch over the wall in right field.
Texas missed out on a chance to extend its lead over Oakland in the American League West. The Rangers remained three games in front of the A's with 19 games to play. Oakland and Texas will meet seven times during the final 10 days of the season.
Not only did Cleveland commit three errors that led to three unearned runs, they squandered chances throughout the game.
The Indians had two men on with either one or two away in the third, fourth and fifth innings and could not come up with a clutch hit. Cleveland then loaded the bases with one out in the eighth and still could not score.
Scott Maine (2-1) got the win by recording the final two outs in the Texas eighth. Chris Perez worked the ninth for his 36th save, but he had an anxious moment when Elvis Andrus tripled with two out. Perez then struck out David Murphy to end the game.