Poythress drove in four runs and is now tied for ninth on Georgia's career home runs (32) and RBI (155) lists.
Chase Hawkins (3-1) picked up the win, allowing just one run on five hits in five innings. Justin Earls, Jeff Walters, Will Harvil and Dean Weaver combined to surrender one run on three hits in the final four innings.
"The beauty of a doubleheader is that you can get a bad taste out of your mouth real quick and we had a bad taste after that first game," Georgia coach David Perno said. "You're only as good as your pitching and defense. It doesn't matter how good you hit, if you can pitch and can't catch, you're not going to win games. We just didn't do that in the first game."
The Bulldogs took a 1-0 lead on an RBI single by Poythress in the first. Winthrop (20-11) answered with a run in the third, then Lewis' grand slam put the Bulldogs ahead 5-1.
Poythress then hit a three-run blast in the fourth, his 14th of the season, to give Georgia an 8-1 lead.
Lewis connected on a solo home run in the sixth, then in the seventh Peter Verdin hit a pinch-hit two-run homer and Levi Hyams hit a two-run blast to make it 13-1. Winthrop added a run in the eighth.
Robert Lake (4-2) took the loss, surrendering seven runs on eight hits in 3.1 innings.
In game one, Winthrop took advantage of four Georgia errors and scored five unearned runs in the 7-4 win. The Eagles took the lead on three unearned runs in the third, and Matt Thielepape led off the fourth inning with a solo home run to give Winthrop a 4-0 edge. Another unearned run in the sixth made it 5-0.
Georgia broke through with a run in the bottom of the seventh and had the bases loaded with two outs when Tyler Mizenko struck out Levi Hyams to get out of the jam.
The Eagles got the run back in the eighth and added another in the ninth to go ahead 7-1. The Bulldogs scored three in the ninth and had the tying run on deck when Poythress grounded out to end the game.
Ken Monteith (3-1) allowed just one earned run on two hits in 6.1 innings to pick up the win. Georgia's Jason Leaver (3-1) allowed three runs—all unearned—on five hits in 2.2 innings to take the loss.
Georgia heads to Kentucky for a three-game series beginning Friday at 6:30 p.m.