Flacco, the brother of Baltimore Ravens quarterback Joe Flacco, got his first hit of the night in the eighth inning, advanced to third on the throw home and scored when GreenJackets 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 put the game-tying or winning run on base in the ninth in each of the other meetings of the five-game series.
"The way the series went, we'd get a lead and then walk the leadoff guy late and let them back in pretty much the whole entire series," Delmarva manager Ryan Minor said. "Especially tonight in the eighth, we got the lead and walked the leadoff guy right away and then you're having flashbacks right there."
No such rally came for Augusta on Saturday.
Delmarva's bullpen shut out the GreenJackets after starter Luis Noel allowed an unearned run in the second inning. Noel earned his first win in front of an announced crowd of 3,856 at Lake Olmstead Stadium.
GreenJackets starter Taylor Rogers (0-3) gave up two earned runs on six hits and struck out a career-high eight batters but still took the loss.
"They jumped all over our walks," Augusta manager Lipso Nava said. "And an error here, an error there. That came back to haunt us."
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 score again. Seven base runners were left stranded, one was picked off first, and three others were caught stealing by catcher Justin Dalles.
Duvall went 0 for 4 to see his 10-game hitting streak come to an end. He was ejected in the eighth after arguing a called third strike.
Schoop extended his hitting streak to 11 games with a double in the third.