Green also had a pair of free throws in overtime and four straight foul shots late in regulation to tie it at 66 for Charlotte (8-6), which won its fourth straight come-from-behind victory late in a game.
Iman Shumpert led the Yellow Jackets (7-6) with a season-high 28 points and Glen Rice Jr. added 18 points.
Shumpert scored all of the Yellow Jackets' points in the first overtime period, including a pair of 3s and a jumper to give them a 74-72 lead.
Braswell then sent it to a second overtime for Charlotte, 74-74, on a layup with 12.9 seconds left in the first overtime.
Georgia Tech had a chance to win it when Rice was fouled with 3.5 seconds left but he missed both, forcing the second extra period.
Rice also missed a pair of free throws late in regulation.
Green hit four straight free throws down the stretch in regulation, including a pair with 1:07 left to tie it at 66 and send it into the first overtime.
Georgia Tech took a 40-37 lead at halftime. The Yellow Jackets took a 34-20 lead earlier on a 9-0 run led by Shumpert, who scored seven of the points and had two steals, including one after which he fed to Rice for a layup that capped the streak with 4:40 left before halftime.
The 49ers, however, made a late first-half run of their own, outscoring the Jackets 17-6 the rest of the way with Barnett scoring eight points, including a pair of 3s.
During Charlotte's previous three wins, the 49ers won by a total of six points. They beat Mercer 64-63 on Thursday, won 57-53 over Wright State on Dec. 22 and beat then-No. 7 Tennessee 49-48 on Dec. 17.