Duncan's homer touched off a five-run fourth inning that also featured a two-run double by Jack Hannahan.
Hannahan later scored on a double by Sizemore, who eventually came home on Orlando Cabrera's double. Davies allowed eight runs on nine hits over 3 1/3 innings.
The Indians swept the three-game series, during which they outscored the Royals 24-8. Kansas City has lost six in a row.
Fausto Carmona (2-3) picked up the victory with a seven-inning effort. He allowed two runs on five hits.
Celebrity Families of 2014 [PHOTOS]