Ashley's shot leading off the eighth inning keyed a two-run rally that gave the Catfish (24-12) a 5-2 lead. A squeeze bunt by Craig Albernaz brought home the inning's other run.
The Catfish, who have won four straight, took a 2-0 lead in the first on Ashley's RBI double to right. He came home on shortstop Chad Lundahl's throwing error.
The Braves (10-25) knotted the game in the second on Chase Fontaine's first homer of the year, a two-run shot.
Matthew Fields singled home Ryan Royster in the fourth with the eventual winning run.
The Catfish have beaten Rome in 8 of 9 meetings this season. The 4-game home stand continues this afternoon.