Asdrubal Cabrera and Ezequiel Cabrera each added two RBI as the Indians avoided a three-game sweep and won for only the second time in six games.
David Huff (2-0) went 5 2/3 innings to claim the victory, giving up just one run on three hits and a walk.
Kansas City's Jake Odorizzi (0-1) took the loss in his first major-league start, allowing three runs on six hits and a walk over 5 1/3 frames.
Adam Moore homered for the Royals.
Celebrity Families of 2014 [PHOTOS]