Buckner (1-0), making his first start for the Angels since being recalled last week from Class AAA Salt Lake City, yielded just two hits. The right-hander walked three and struck out two.
A former Royals draft choice, the 29-year-old last notched a win three years ago for the Arizona Diamondbacks.
Dane De La Rosa, Scott Downs, Robert Coello and Michael Kohn each tossed a scoreless inning in relief to complete the shutout for the Angels, who have won seven straight.
Royals starter Jeremy Guthrie (5-3) was roughed up for seven runs (six earned) on five hits over 7 1/2 innings.