The GreenJackets (28-18) managed only one run on four hits until an unsuccessful rally in the ninth produced three runs. They dropped their second game in as many nights to a team who entered the series with the worst record in the South Atlantic League.
"We're a better club than this," third baseman Chris Dominguez said. "We've faced some good arms this series, but we have to keep playing and not give up."
Rome (19-26) knocked around Heston for six runs on 11 hits before he was pulled with two outs in the fifth. The Braves finished with 17 hits, one short of their season high.
"He's getting his pitches up and out over the plate, and he's not throwing his breaking ball for strikes," Augusta manager Dave Machemer said of Heston.
Rome starter Cory Rasmus (2-3) held Augusta scoreless, with the exception of a home run by Luke Anders in the fourth inning.
Anders' team-leading seventh homer of the season cleared the scoreboard in right-center field at Lake Olmstead Stadium and broke up Rasmus' no-hit bid.
The GreenJackets, who trailed 8-1 entering the ninth, put together a rally with three consecutive singles by Nick Liles, Tommy Joseph and Hector Sanchez. Sharlon Schoop immediately followed with a fly ball to center field that appeared to be good for a sacrifice fly, but the ball hit off the side of Rome center fielder Kyle Rose's glove to score one run and put runners at second and third with one out.
Ryan Lollis grounded to first to score another run, but Evan Crawford struck out on three pitches to end the game.
Joseph, who entered the game in a 3 for 35 slump, went 2 for 4.
Chris Curley led the Rome offensive charge with a career-high four hits. Cory Harrilchak and L.V. Ware drove in two runs apiece, as two of the six Braves batters who turned in multi-hit nights.
The GreenJackets' lead in the SAL Southern Division standings dropped to one game over the Savannah Sand Gnats.