Greinke (12-8) gave up five hits and a walk over eight innings. The previous Kansas City record for most strikeouts in a single game had been 14, set 21 years ago by Mark Gubicza.
Greinke's previous strikeout high had been 11.
Mitch Maier backed up Greinke with his first major league homer, a two-run shot in the seventh that gave the Royals a 6-1 lead. Kansas City snapped a five-game losing streak.
Cleveland had won its previous three games.
Justin Masterson (4-5) took the loss, giving up four runs on seven hits with two walks in six innings.
Copyright 2009 by United Press International.
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