Pinch-hitter Chris Herrmann began the inning by coaxing a walk from Craig Stammen (3-2) and went to second on a Jamey Carroll sacrifice.
After Joe Mauer was intentionally walked, Doumit lined a pitch into center field for a solid hit, chasing home Herrmann with the go-ahead tally.
Josh Roenicke (2-1) earned the victory with scoreless top of the 11th and Glen Perkins logged his 13th save for Minnesota, which won the first of a three-game series after Friday's scheduled opener was rained out.
That game will made up as part of a Sunday doubleheader.
Mauer finished 3-for-5 with a solo homer.