Aoki also had an RBI on a third-inning single, but it was his two-strike bunt base hit to lead off the Royals' half of the eighth with the game knotted at 1-1 that turned into the game's pivotal moment.
Alcides Escobar followed with a sharp double off Chicago starter Jose Quintana for his second hit of the night, which set up Billy Butler's tie-breaking sacrifice fly that brought home Aoki without a throw.
Quintana (2-4), who entered the eighth having retired 10 straight Royals, was charged with all three runs over 7 1/3 innings in a hard-luck defeat.
Danny Valencia also drove in a run to help Kansas City avoid a sweep of the three-game series.
Swim Week Miami Beach 2014 [PHOTOS]