The Royals (10-11) managed just four hits and scored an unearned run in the loss. They dropped three of four in the series, and have lost four of five. Despite the recent drought, Kansas City's just two games back of Detroit for the top spot in the American League Central.
Bruce Chen was doing his best to match Kluber through the first four innings. Prior to the fifth, the Royals's starter had allowed just one hit and no walks, but the Indians wore him down.
Cleveland scored five runs on five hits off Chen to break the game open. Daniel Murphy and Asdrubal Cabrera delivered the biggest blows driving in two runs apiece.
Kansas City's only offense came in the seventh inning on an error by Nick Swisher. Alcides Escobar crossed the plate for the Royals' lone run.
Chen drops to 1-2 on the season after completing just 4 1/3 innings. Kluber improved to 2-2 tossing his first complete game.
The Royals road trip continues this weekend as they open up a three-game set in Baltimore. Yordano Ventura will get the start for Kansas City on Friday.