Heyward followed with a double, scoring Johnson with the tying run, and Simmons also scored for a 6-5 lead when outfielder Jason Kubel threw wildly back to the infield.
Brian McCann, Dan Uggla, B.J. Upton and Chris Johnson each added an RBI before the inning was through as the National League East-leading Braves clinched their three-game series from NL West-leading Arizona.
Jordan Walden (3-1) was credited with the victory despite yielding a go-ahead homer to Miguel Montero in the top of the eighth.
A.J. Pollock cracked a two-run shot for the Diamondbacks.