Greinke (12-10) walked one batter, fanned seven, and threw 61 of 97 pitches for strikes. His effort led to Kansas City's seventh straight victory.
Mike Aviles' only hit, a three-run third inning homer, started the rout. Alex Gordon's only hit was a two-run sixth inning double. A run scored on Jared Wells' fourth inning wild pitch.
The Royals' Mark Teahen led a 13-hit attack with three hits.
Mitch Maier had two hits and eight Kansas City players had one hit each.
Three players had one hit apiece in the Mariners' eighth loss in a row.
Losing pitcher Ryan Feierabend (1-4) was torched for eight earned runs and six hits with three walks in 3 1/3 innings.