Adam Jones reached base on third baseman Ty Wigginton's error with one out and Wieters followed by belting a pitch from Joe Savery (0-1) off the right field wall, chasing Jones home with the winning run.
The victory gave the Orioles a club-record nine straight extra-inning triumphs and handed them the rubber match of a three-game series at Camden Yards.
Weiters finished 3-for-4 and Scott Tolleson belted a three-run homer for Baltimore, which remained one game out of first place in the American League East.
Darren O'Day (4-0) earned the win with a scoreless top of the 10th.
Hunter Pence went 2-for-4 with a pair of RBI for the Phillies.
Notable deaths of 2014 [PHOTOS]
Michael Sam cut from the Dallas Cowboys