Trailing 4-3 entering the eighth, Minnesota began a comeback with Joe Mauer stroking a one-out single off A's closer Ryan Cook (1-1) and stealing second base. Josh Willingham drew a walk and Justin Morneau followed with a bloop double down the left-field line, chasing Mauer home with the tying run and sending Willingham to third.
Doumit's fly ball to left then pushed across the decisive run for Minnesota, which broke a five-game losing streak in the opener of a three-game set.
Alex Burnett (2-0) got the victory after notching the final two outs in the eighth and Matt Capps picked up his 10th save despite surrendering two walks in the ninth.
Josh Reddick hit a triple and a solo homer and Kila Ka'aihue also went deep for Oakland.