The division champion Braves pulled even with idle St. Louis for the best record in the National League.
Atlanta produced six of their 12 hits in the first inning to take control and David Hale (1-0) was able to coast to the first victory of his major league career.
Hale struck out nine in five scoreless innings last Friday in his big league debut against San Diego. They were the most strikeouts ever for an Atlanta pitcher in his first career start.
Hale gave up a run on seven hits in six innings against the Phillies.
After Heyward hit his 14th home run of the season, Elliot Johnson kept the first inning going with a single and he scored on a double by Evan Gattis. Chris Johnson doubled home Gattis and Johnson scored the final run of the inning on a single by Jordan Schafer.
A two-run double by Gattis in the second completed the scoring.
Cloyd (2-7) was charged with all seven runs while retiring only three batters.