The Braves streak is the longest in the majors this year and has turned the National League East pennant race into an apparent formality. Atlanta leads second-place Washington by 15 1/2 games.
It was the Nationals' fourth straight loss.
Both sides scored in the seventh to bring about a 3-3 tie, but Atlanta took charge in the eighth by sending eight batters to the plate.
Washington reliever Ryan Mattheus (0-1) retired the first two men in the inning, but Andrelton Simmons then doubled and Joey Terdoslavich drew a walk. Ian Krol relieved Mattheus and Heyward greeted him with a single that brought home Simmons.
Justin Upton followed with a double that drove in two more runs.
The big eighth inning made a winner out of Atlanta starter Kris Medlen (9-10), who worked seven frames while giving up three runs on just three hits. Craig Kimbrel loaded the bases in the ninth, but got Wilson Ramos on a liner to right that ended the game. It was Kimbrel's 36th save.
Two of Washington's runs came on a fourth-inning homer by Jayson Werth.