With the scored tied 6-6, Adam Jones led off the 12th with a two-bagger off of Alfredo Aceves (2-10) and one out later Thome blasted a pitch into the right field corner, where it one-hopped the fence for a ground-rule double.
Endy Chavez and Manny Machado (3-for-6, two RBI) later added run-scoring singles and the American League wild card-leading Orioles won their 16th straight extra-inning game.
Baltimore entered the contest trailing the first-place New York Yankees by one game in the AL East.
Jones and Ryan Flaherty each drove in a pair of runs as the Orioles clinched their three-game series.
Tommy Hunter (6-8) pitched a scoreless 11th for the victory and Jim Johnson racked up his 47th save for Baltimore.
Danny Valencia socked a two-run homer and Ryan Lavarnway knocked in a pair for the Red Sox.
Notable deaths of 2014 [PHOTOS]
Michael Sam cut from the Dallas Cowboys