Baynes scored on a pass from Patty Mills and the Hawks couldn't get off a last-second shot.
Tony Parker scored 17 points, including 14 in the third quarter, to lead the Spurs.
Kyle Korver led Atlanta with 26 points.
The Hawks trailed by 14 before rallying in the fourth quarter. Korver's fifth 3-pointer tied the game at 97-97.
San Antonio returns 12 players, including all five starters, from the team that lost to Miami in the NBA Finals.
Jeff Teague had 15 points for Atlanta, including a 3-pointer to tie the game at 104-all. DeMarre Carroll had 14 points and Al Horford had 13 points and 11 rebounds. Paul Millsap had 10 points and 11 rebounds. Jared Cunningham, competing for a spot as a backup guard, had 11 points, including two crowd-pleasing jams.
The Spurs led 44-41 at halftime before outscoring the Hawks 38-26 in the third quarter.
The game provided Spurs coach Gregg Popovich time to visit his former longtime assistant, first-year Hawks coach Mike Budenholzer. The Hawks have another Spurs connection in second-year general manager Danny Ferry, who came to Atlanta after two years as San Antonio's vice president of basketball operations.
Budenholzer spent 19 years with the Spurs, including 17 as Popovich's assistant.
"I feel like I'm lost half the time when I turn to my right and my left and he's not there, suggesting to me who I should have in the game and that I should be changing the pick-and-roll defense," Popovich said. "All that kind of stuff. I'm a little bit more free now, but I'm not as smart."