The Braves scored three times in the bottom of the first to build a quick 3-0 lead against Mets starting and losing pitcher Pedro Martinez (5-6).
Four players had one RBI and Josh Anderson had three of the nine hits for the Braves, who stopped a four-game losing streak.
Campillo (8-7) walked two batters, struck out two, and won for the first time since Aug. 8 at Arizona.
Martinez knocked in both New York runs with a fifth-inning double.
David Wright had two of the six hits for the Mets, who had won three in a row.
With the loss, the Mets fell one-half game behind Philadelphia in the National League East.
The Phillies ended the Florida Marlins' nine-game winning streak with a 3-2 victory.
2014: The Year in Music [PHOTOS]