Drew's three-run shot in the fourth inning staked the Red Sox to a 4-0 lead and his two-run, inside-the-park homer in the sixth gave the Bosox a 6-1 cushion.
The Boston shortstop finished with three hits and scored three times while Shane Victorino socked a solo homer and added an RBI groundout as the Red Sox evened their three-game series.
Ryan Dempster (6-8) went 5 1/3 innings to gain the victory, surrendering two runs on six hits. The right-hander walked two and struck out four as second-place Boston remained a half-game behind American League East-leading Tampa Bay.
Orioles starter Scott Feldman (2-2) was saddled with the defeat after yielding four runs on six hits over five innings.