With the game in a scoreless tie, Justin Upton and Freddie Freeman hit consecutive doubles off Henderson Alvarez to lead off the fourth and Gattis followed with another two-bagger, sending a pitch down the third-base line for two RBI.
Chris Johnson and Elliot Johnson each added an RBI in the five-run outburst, which handed the National League East-leading Braves a win in the first of a four-game series.
Kris Medlen (13-12) earned the victory with 6 1/3 solid innings, allowing two runs on six hits. He walked two and struck out six.
Alvarez (3-4) was roughed up for all five Atlanta runs over four full innings, surrendering five hits and three walks.
Ed Lucas drove in both Miami runs and Chris Coghlan went 4-for-4 in the loss.