Georgia (25-4, 8-3 SEC) led 10-3 going to the eighth, but LSU (22-8, 7-4 SEC) scored three in the eighth and two in the ninth and had the tying run at the plate with two outs when Dean Weaver got Mikie Mahtook to pop up to end the game.
Matt Cerione, Colby May and Levi Hyams each added two hits and May had three RBI.
Alex McRee (3-0) threw a career-long six innings and surrendered three runs-two earned-on seven hits to pick up the win.
"We had a much better approach today and got some key hits," Georgia coach David Perno said. "We struck early and answered, and we played more to our identity. We are moving in the right direction and today was a huge step for us."
Jared Mitchell gave LSU a 1-0 lead in the top of the first with a solo home run, then Georgia responded with four runs in the bottom of the inning. After a leadoff single by Cerione and a walk to Poythress, Massanari's home run gave Georgia a 3-1 lead. Lyle Allen then walked, moved to third on a double by Lewis and scored on May's sacrifice fly to make it 4-1.
LSU scored on a passed ball in the third, then the Bulldogs came back with a pair of runs. Massanari singled then scored on another double by Lewis, and May followed with a base hit to score Lewis and make it 6-2.
Ryan Schimpf hit a solo home run in the fourth to pull LSU to within three, then Georgia posted single runs in the fifth and sixth. A two-run seventh extended the lead to 10-3 before LSU rallied in the final two innings.
Austin Ross (4-3) took the loss for the Tigers, allowing six runs on nine hits in three innings.
Georgia and LSU will play the rubber game of the series Sunday at noon. Sophomore righty Justin Grimm (2-1, 4.11) will get the start for the Bulldogs, and LSU will counter with right-hander Louis Coleman (6-1, 1.38).