The Orioles let an early three-run lead get away and the Red Sox forged a 4-4 tie in the seventh on a sacrifice fly from Stephen Drew and an RBI, fielder's choice grounder off the bat of Jacoby Ellsbury.
That was all the scoring until Baltimore got to Alex Wilson (1-1) in the 13th.
Nate McLough struck out to open the inning and Manny Machado grounded out to third.
Markakis then walked on a 3-2 pitch and moved to second on a single by Adam Jones. That brought up Davis, who sent a 1-1 offering to left field for the winning hit.
T.J. McFarland (1-0) picked up his first major league victory in his 15th career relief appearance.
McFarland, the seventh Baltimore pitcher to see action in the game, worked a perfect top of the 13th to get the win.