Guillen's fifth-inning blast off the Twins' Kevin Slowey (2-2) gave the Royals the lead for good at 3-2 and Mitch Maier's RBI triple the next inning got Kansas City an insurance run as they avoided a three-game sweep.
Yuniesky Betancourt chipped in with an RBI while Bannister (1-1) hung on for the decision despite allowing nine hits and two walks over 6 1/3 innings.
Justin Morneau socked a solo homer and drove in two, while Joe Mauer got a pair of hits after going 5-for-6 in Saturday's 12-inning Twins win over the Royals.
Slowey (2-2) went 5 1/3 innings, yielding four runs on eight hits.
Celebrity Families of 2014 [PHOTOS]