Aybar's third hit snapped a 1-1 tie with one out in the top of the 11th.
The winner was Kevin Jepsen (5-3).
Closer Brian Fuentes escaped an 11th-inning jam to record his 39th save.
Starter John Lackey surrendered a run and six hits in nine innings.
Kansas City's lone run scored on Billy Butler's first-inning single.
He had two of the seven hits for the Royals, who have lost three straight and four of five.
The losing pitcher was Yasuhiko Yabuta (0-1).
Starter Zack Greinke gave up a run and eight hits in eight innings with two walks and eight strikeouts.