Adam Duvall's throwing error extended the third inning long enough for Rome's Chris Garcia to hit a three-run homer.
All four runs in the inning were unearned and made the difference in Augusta's first loss of the four-game series.
"That was the game there. Just one error and one pitch," GreenJackets manager Lipso Nava said. "It's an everyday thing. It's either pitching or hitting or today we made two errors. We've been playing really good defense, but tonight we made two errors and one of them cost us some runs and some extra pitches for our pitchers."
Rome added another run in the fifth on a solo homer from Robert Hefflinger to give the Braves their second win in six games.
Braves starter Caleb Brewer got his first victory after losing his two previous outings. The 22-year-old right-hander allowed two runs on seven hits and two walks.
Reliever Ryan Weber retired six of the seven batters he faced over two scoreless innings, and Tyler Hess took care of the ninth for his first save.
Just one of the five runs allowed by GreenJackets starter Shawn Sanford (1-3) was earned. He allowed seven hits, walked one and struck out three on a night when he was pleased with the movement on his pitches but displeased with the outcome.
"There was an error, but the fact of the matter is I gave up a home run," he said. "Instead of a one-spot, I didn't battle hard enough and gave up a four-spot. If I do my job, we hold it to one run and we win the ballgame."
Three Augusta relievers held Rome scoreless over the final four innings.
Edward Concepcion highlighted the bullpen effort by striking out all three batters he faced in a perfect ninth.
But the GreenJackets offense stranded eight runners, including one in scoring position in the ninth. Augusta managed just one hit -- Chris Lofton's infield single -- over the final three innings.
Bobby Haney went 2 for 3, including an RBI double in the fifth. Duval hit his sixth home run on a solo shot in the fourth.
Garcia, who hit his second homer of the series, went 2 for 4 with three RBI.