With the game scoreless in the seventh inning, Adam Jones and Matt Wieters led off with singles and Davis then powered a pitch from Tigers starter Rick Porcello (9-8) the opposite way into the Detroit bullpen for a three-run shot.
Britton (2-1) threw seven shutout frames, scattering six hits while working around three walks and striking out five.
Pedro Strop surrendered two runs in the eighth but Jim Johnson closed the door in the ninth for his 36th save as the Orioles evened their three-game series.
Baltimore, Detroit and the Tampa Bay Rays came into the contest tied for the two American League wild card spots.
Porcello ended up allowing the three runs on seven hits in six innings.
Jhonny Peralta drove in both Detroit runs.