Heyward ended with four RBI and Martin Prado drove in three runs on two hits for the Braves, who took the first of a four-game series at Turner Field and won their third straight contest.
Omar Infante scored three times as Atlanta, which entered the game with a two-game lead over the second-place Phillies in the National League East, raised its home record to 47-18.
Jair Jurrjens (6-4) allowed three runs on six hits over 5 2/3 innings, fanning eight and surviving six walks for the win.
Mets starter Pat Misch (0-4) was rocked for five runs on eight hits in just three innings.
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