The Orioles were within one out of a victory in the bottom of the ninth when closer Jim Johnson (1-0) surrendered a score-tying, two-run homer to Boston's Jarrod Saltalamacchia.
But the O's came back in the next half-inning when Mark Reynolds drew a leadoff walk off Alfredo Aceves (0-3), went to second on Steve Pearce's sacrifice and scored on Paulino's hit to shallow center field.
Paulino advanced to third on Ryan Flaherty's single and came home with an insurance run on an Endy Chavez hit, and Johnson hurled a 1-2-3 bottom of the 10th to secure Baltimore's sixth straight victory at Fenway Park.
Chavez finished 3-for-5 with two runs scored and Pearce added two RBI for the Orioles, who improved to 7-2 in extra-inning games.
Saltalamacchia ended with three hits and Mike Aviles went 3-for-5 with an RBI for the Red Sox, who dropped the opener of a three-game series against the Orioles.