Austin Jackson had two hits and three RBIs for the Indians, while Michael Brantley, Jose Ramirez and Carlos Santana each had three hits, as the Tribe pounded out 18 hits.
Royals left-hander Danny Duffy lasted only four-plus innings, giving up nine hits and six runs.
Rookie Jorge Bonifacio belted his seventh home run to account for the Royals' lone tally.
MOMENTS THAT MATTERED
Turning point: The Royals had crept within 3-1 when Brantley led off the Indians' fifth with a harmless looking ground ball toward first baseman Eric Hosmer.