The White Sox handed the Rangers their first series defeat of the season, winning two of three.
Adrian Beltre put Texas in front with a home run in the second and the Rangers were still ahead when rookie starter Justin Grimm (2-1) went out to work the sixth.
Grimm retired the first two batters in the inning, but then gave up singles to Conor Gillaspie and Alexei Ramirez before Flowers launched his homer to left.
That made a winner out of Chicago starter Hector Santiago (1-1), who allowed the lone Texas run on two hits in 5 1/3 innings.
Texas put two runners on base in both the eighth and ninth innings, but relievers Jesse Crain and Addison Reed worked their way out of trouble. Reed struck out both Mitch Moreland and Leonys Martin in the ninth with the tying runners on to earn his 10th save.