Santiago (3-1), who had made 38 career relief appearances previously, made the most of his first start, holding the Twins to one run on three hits. The left-hander walked three and struck out six.
Five White Sox relievers combined to yield one run over three frames and Addison Reed finished up with his 25th save as the White Sox took the opener of a three-game series and reclaimed first place in the American League Central.
The victory, combined with Detroit's loss to Cleveland earlier Monday, gave the Chisox a one-game lead over the Tigers.
Gordon Beckham smacked a two-run homer in the second inning and added an RBI single in the eighth for the White Sox.
Twins starter Samuel Deduno (5-3) allowed two runs on six hits in a six-inning effort.
Jamey Carroll went 2-for-3 with a solo homer for Minnesota.