Mike Scott added 19 points while Kyle Korver had 18 for Atlanta (4-3), which shot 52 percent from the field.
Tony Parker had 17 points and seven assists, Cory Joseph added 14 points and Boris Diaw had 12 points for San Antonio (2-4). Tim Duncan played just 13 minutes and the Spurs were missing Kawhi Leonard (eye infection) and Tiago Splitter (calf).
There were a combined 50 fouls and 33 turnovers in a sloppy game.
The Hawks scored 42 points in the third, the most the Spurs have allowed in a quarter during the preseason.
Atlanta took a 64-54 lead on Teague's jumper with 8:43 left in the third quarter. After the Spurs pulled within 69-64 on a three-point play by Parker, the Hawks went on a 21-6 run to take control.
Diaw's insertion into the starting lineup had helped ignite an offense that has struggled early in the first two home games of the preseason.
The Spurs opened the game shooting 3 for 4, including a 20-foot jumper and 3-pointer from Diaw off feeds from Marco Belinelli.
Atlanta missed its first five shots, but rallied to shoot 40 for 77.
Teague had 11 points in the second quarter, including an 11-foot floating jumper to tie the game at 46 with 1:26 left in the first half.
Hawks: Atlanta has lost 16 straight regular season games in San Antonio, with its previous victory a 109-89 decision on Feb. 15, 1997. The Hawks play the Spurs Nov. 5 in San Antonio. ... The Hawks averaged 111.3 points in their final three preseason games. ... Atlanta had four players average double figures in the preseason, with Millsap averaging a team-high 13 points six games.
Spurs: Duncan is the only player to post at least 10 rebounds in a preseason game, posting that against Alba Berlin on Oct. 8. ... Rookie Kyle Anderson did not play in the first half, but finished with 10 points in 18 minutes. Anderson came in averaging 28.6 minutes per game, trailing only Parker, Duncan and Leonard on the team. ... San Antonio has allowed 100-plus points in four of its last five preseason games, with Sacramento scoring 99 in a seven-point loss Monday.
Atlanta visits Toronto on Friday to open its regular season.
San Antonio visits Houston on Friday to close its preseason.