Doumit's first homer in the second inning opened the scoring and put Minnesota ahead to stay, while his two-run single in the next frame highlighted a four-run Twins outburst that iced the contest.
The designated hitter added another bases-empty blast in the sixth as the Twins went on to claim the rubber match of a three-game series at Kauffman Stadium.
Brian Dozier supplied a two-run double for the Twins.
Samuel Deduno (1-0), a 29-year-old career minor-leaguer, earned his first big-league victory, holding the Royals to one run on six hits over 6 1/3 frames. The right-hander walked three and struck out four.
Billy Butler socked a two-run homer and Yuniesky Betancourt added a solo shot for Kansas City.