Raburn's three-run shot capped a six-run fifth inning and his solo blast in the eighth provided the final margin for victory for the Indians, who banged out 14 hits en route to a split of their four-game road series.
Jason Kipnis also homered and Asdrubel Cabrera went 2-for-3 with two RBI in the one-sided victory.
Jimenez (1-2) recorded his first victory of the season in his fifth start, dominating the Royals on three hits. The right-hander walked two and struck out four.
Kansas City starter Wade Davis (2-2) was pounded for eight runs on 12 hits, lasting just 4 2/3 frames.