Hosmer's base-on-balls from Scott Kazmir in the seventh inning gave the Royals a 3-0 lead and proved to be the difference as Kansas City took the opener of a four-game series.
Salvador Perez socked a two-run homer and Ervin Santana (3-1) corralled the victory with seven shutout frames, scattering seven hits while striking out five with no walks.
Kazmir (0-1) was touched for two runs on five hits over five full innings.
Michael Brantley delivered a two-run triple in the ninth for the Indians but was stranded at third as Greg Holland picked up his sixth save.
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