Kluber (7-5) struck out eight and gave up just three hits with three walks while winning for the first time in four starts.
Cody Allen allowed two hits in the ninth, but struck out the side to record his second save in 40 relief appearances this season.
Bruce Chen, who had been relegated to the bull pen by Kansas City this year, made his first start of the campaign and held Cleveland scoreless during a six-inning effort.
Tim Collins (2-4) replaced Chen to start the second and immediately gave up singles to Jason Kipnis and Nick Swisher.
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