Yu Darvish (14-9) carried a no-hitter until two were out in the sixth inning, eventually allowing three hits over seven frames. The Japanese right-hander walked one and struck out six.
American League West-leading Texas took the opener of a four-game series.
Royals starter Bruce Chen (10-11) was rocked for six runs on six hits over six innings.
Tony Abreu went 2-for-3 with three RBI for Kansas City.
2014: NFL Cheerleaders [PHOTOS]
Michael Sam cut from the Dallas Cowboys