Minnesota owned a 5-4 lead heading into the bottom of the eighth, but the Orioles loaded the bases with one out and Adam Jones singled in a run.
That brought up Davis, who homered on the first pitch he saw from reliever Tyler Robertson.
Davis had two hits while Jones went 3-for-5 with two RBI and three runs scored in the Orioles' home opener.
The Twins had built their lead thanks in part to three RBI from Brian Dozier.
Luis Ayala (1-0) worked 1 2/3 innings of scoreless relief to get the victory. The loss was charged to Casey Fien (0-1), who loaded the bases in the eighth before Robertson surrendered the blast by Davis.