Kluber (5-4) turned in his second straight strong start for the Indians following eight innings of one-run ball in his previous outing against Texas.
The right-hander scattered seven hits and didn't issue a walk while striking out eight in the 114-pitch gem.
Vinnie Pestano provided a scoreless ninth for his first save of the season as Cleveland took the rubber match of a three-game interleague series.
Carlos Santana and Jason Kipnis each knocked in a run for the Tribe.
Nationals starter Stephen Strasberg (3-6) pitched well in his return from a stint on the disabled list with a strained lateral muscle. He held the Indians to one run on one hit, walking four and striking out four.
Celebrity Families of 2014 [PHOTOS]