The Braves had been blanked over eight innings and came into the ninth trailing 1-0, but mounted a rally against Mets closer Bobby Parnell (5-5).
Brian McCann led off the frame with a single and pinch-runner Jose Constanza went to second on an Evan Gattis single. After Dan Uggla's grounder forced Constanza at third, both he and Gattis advanced on a passed ball by Mets catcher John Buck.
Gattis scored the equalizer on Chris Johnson's infield single and pinch-hitter Johnson followed with a solid hit to center, chasing home the decisive run.
Braves closer Craig Kimbrel, however, struggled in the bottom of the ninth, allowing New York to put a runner on second with two outs.
Pinch-hitter Justin Turner lined a shot to center that Heyward dove to snare, ending the contest and handing a victory to reliever Jordan Walden (4-1).
Julio Teheran tossed six innings of one-run ball for National League East-leading Atlanta, which won the opener of a three-game series.