Davis' three-run blast in the first inning staked the Orioles to a quick 4-0 lead and his two-run shot in the sixth capped the scoring for Baltimore, which clinched its three-game series against the Yankees.
Ryan Flaherty chipped in a three-run homer and Chris Dickerson drove in two runs for the second-place O's, who pulled within 2 1/2 games of American League East-leading Boston with their third straight win overall.
Zach Britton (2-2) logged the victory, yielding two runs (one earned) on six hits over 5 2/3 innings. The left-hander walked three and struck out one.
Yankees starter David Phelps (5-5) lasted just 2 1/3 frames, coughing up nine runs on nine hits.
Brett Gardner and Zoilo Almonte each drove in a run for New York, which fell 5 1/2 games out of first place with a fourth consecutive defeat.