Bannister (6-3) was sharp in posting a win in his fifth straight start, holding the Tigers to two runs on five hits over 7 1/3 innings, issuing only one walk and fanning five.
Bannister raised his career record against Detroit to 5-2 with the win.
Guillen's blast in the first inning staked the Royals to a quick lead. Yuniesky Betancourt doubled and drove in a pair and Jason Kendall collected two hits and an RBI to help give Kansas City a win in the rubber match of a three-game set.
Jeremy Bonderman (2-4) absorbed the loss for the Tigers, yielding seven runs on 11 hits over 5 2/3 innings.