Flacco, the brother of Baltimore Ravens quarterback Joe Flacco, got his first hit of the night with the bases-clearing double in the eighth, advanced to third on the throw home and scored when catcher Dan Burkhart threw the ball over the head of third baseman Adam Duvall.
The four-run eighth put the game out of reach for the GreenJackets, who had managed to put the game-tying or winning run on base in the ninth in each of the last four games.
No such rally came Saturday. Delmarva's bullpen shutout Augusta after starter Luis Noel allowed a lone unearned run in the second inning on his way to his first win.
GreenJackets starter Taylor Rogers (0-3) gave up two earned runs on six hits and struck out a career-high eight batters but still got the loss.
Augusta's Rafael Rodriguez extended his hitting streak to eight games with a single in the second and later scored on an error by shortstop Jonathan Schoop.
The GreenJackets failed to manufacture any other runs. Seven base runners were left stranded, one was picked off first and three others were caught stealing by catcher Justin Dalles.
Duvall saw his 10-game hitting streak come to an end with an 0 for 4 night. He was ejected in the bottom of the eighth after arguing a called third strike.
Schoop extended his hitting streak to 11 games with a double in the third.