Smith (1-1) collected his initial victory in his second career start, yielding two runs on four hits over six innings. He walked two and struck out five.
Moustakas clubbed a two-run homer in the first inning that put Kansas City ahead to stay and drove in two more with a second-inning single, capping a five-run Royals outburst.
Humberto Quintero added two RBI for Kansas City, which evened its three-game series at Progressive Field.
Indians starter Justin Masterson (2-4) was tagged for eight runs (seven earned) on nine hits over six full frames.