Doumit led off the 10th with a two-bagger into the left-field corner against Hector Santiago (0-1).
The miscue made a winner of Casey Fien (1-1), who got the final out of the eighth. Glen Perkins hurled a scoreless ninth for his fourth save as the Twins took the opener of a rain-shortened two-game series.
Minnesota starter Vance Worley held the White Sox to one run on five hits and two walks through seven innings, striking out seven.
Chicago's Jake Peavy also pitched well in a no-decision, yielding a single run on six hits with fanning nine over seven frames.
Alejandro De Aza smacked a solo homer for the Chisox.