Morneau's two-run single capped a four-run fourth inning that enabled Minnesota to erase a 4-2 deficit and Doumit's blast in the sixth iced the Twins' fifth victory in seven games.
Joe Mauer contributed three hits, scored three times and knocked in a run while Brian Dozier helped out with a two-run double as Minnesota evened its three-game series against the Orioles the night after Baltimore rallied from a 6-0 deficit to win 9-6.
Vance Worley (1-4) garnered his first win as a Twin in his eighth start of the season despite surrendering five runs on 11 hits over 5 1/3 innings.
O's starter Steve Johnson (0-1), making his initial start of 2013 after starting the season on the disabled list, was roughed up for six runs on seven hits over four full frames.
Manny Machado went 3-for-5 with an RBI and Matt Wieters drove in two runs, including one on a solo homer, for Baltimore.