Greinke (3-0) prevailed in a matchup of aces who started the day with the two lowest earned-run averages in the American League - Greinke at 0.00 and Texas’ Kevin Millwood at 0.64.
Greinke has thrown 34 consecutive scoreless innings, including 20 over his three outings this year.
He struck out a season-high 10 and didn’t issue a walk while getting his fourth career complete game in 106 starts.
With a runner on third in the ninth, Greinke struck out Chris Davis looking to end the game, wrapping up the second shutout in the AL this year.
Millwood (1-1) also turned in a strong effort, allowing two runs and five hits in nine innings for his 18th career complete game. He struck out three and walked three.
Kansas City has won five of six. Texas has lost seven of eight.
The Royals took the lead in the sixth on Billy Butler’s two-out RBI double. Coco Crisp scored after reaching on a walk and advancing to third on consecutive grounders.
Miguel Olivo’s two-out solo homer in the seventh extended Kansas City’s lead to 2-0.
