Dempster gave up two runs off seven hits in seven innings and struck out eight.
Rangers third baseman Adrian Beltre left the game in the fourth inning due to a left shoulder sprain, apparently suffered in the second inning when diving for a Russ Canzler ground ball. He will have an MRI as a precautionary measure, but no results are expected on Wednesday night.
Beltre hit his 32nd home run of the season in the first inning, before leaving the game. The two-run shot came after Josh Hamilton scored on a fielder's choice to open the scoring.
Hamilton hit a solo home run in the third and the Rangers held a 4-0 lead heading into the fourth inning.
The Indians' Carlos Santana scored on a fielder's choice in the top of the fourth, then Geovany Soto hit a sac fly in the bottom of the inning to make it 5-1. A Santana sac fly in the fifth made it 5-2 and that's how it would stay.
Hamilton left the game in the eighth inning with left knee soreness, replaced by Craig Gentry. Mike Olt, who replaced Beltre, left in the seventh. He has been dealing with plantar fasciitis in his right foot.
Jeanmar Gomez (5-8) took the loss for Cleveland. He gave up five runs and six hits in four innings.
Joe Nathan converted his 35th save of the season and his 31st in a row.
The series concludes on Thursday. First pitch is scheduled for 7:05 p.m. CT.
Head over to Lone Star Ball for more Rangers coverage, and be sure to check out Let's Go Tribe for more on the Indians. Also visit Baseball Nation for even more news and analysis from around the league.