Stults (3-2) controlled the Braves for 7 2/3 innings, giving up just the one run and allowing all five of their hits. The left-hander, making his second start since straining a back muscle in early June, didn't issue a walk and fanned three.
Luke Gregerson recorded the final out of the eighth inning and Dale Thayer collected his sixth save with a 1-2-3 ninth for San Diego, which took the first of a four-game set.
Headley's RBI single in the fifth gave the Padres a 2-0 lead and his solo homer two frames later capped San Diego's scoring.
Mike Minor (6-9) took the loss for the Braves, coughing up three runs on nine hits in a six-inning effort.
Martin Prado got the only RBI for Atlanta, which came into the contest with a 4 1/2-game lead for a National League wild card spot.