Clinging to 2-1 lead, Baltimore exploded for five runs in the sixth to take control of the contest.
The Orioles put J.J. Hardy and Adam Jones aboard with one out and Matt Wieters bounced a pitch off the plate for an infield hit, sending Hardy home for a two-run lead.
Reynolds had belted a solo homer in the previous frame while Omar Quintanilla added a bases-empty clout for the Orioles, who came into the contest with a half-game lead for an American League wild-card spot.
Wei-Yin Chen (11-7) tossed six innings of one-run ball to earn the victory, working around nine hits while walking one and striking out four.
Beckett finished by allowing six runs on six hits over 5 1/3 innings.
Carl Crawford went 2-for-4 with an RBI for the Red Sox.