After giving up a one-out single in the second inning to Matthew Weaver, Stoneburner did not allow another baserunner until L.V. Ware singled leading off the ninth. The 22-year-old right-hander was relieved by Eric Flannery, who was trying to nail down his fifth save.
Ware stole second and continued to third on an errant throw by catcher Francisco Arcia. Flannery hit Kyle Rose and gave up an RBI bunt single to Jordan Kreke but retired Cory Harrilchak and pinch-hitter Christian Bethancourt on ground balls.
Mycal Jones followed with another grounder to first base but beat it out as Rose and Kreke scored to force extra innings.
Stoneburner, a 14th-round pick by the Yankees in last year's Draft, remained winless in five professional appearances, including four this season. He was charged with one run on three hits, matched a career high with eight strikeouts and did not walk a batter.
Since surrendering four runs -- three earned -- over 4 1/3 innings in his season debut on April 9, the Clemson University product has yielded only three earned runs in 20 2/3 innings to lower his ERA to 2.16.
Charleston won it in the 12th when Raymond Kruml and Jimmy Paredes singled and Luke Murton delivered a walk-off double. The victory was the RiverDogs' 500th at Joseph P. Riley Jr. Park, which opened in 1997.
Julio Teheran started for Rome and struck out seven in 4 2/3 innings. He gave up two unearned runs on two hits.