Atlanta swept the three-game series and has won four in a row overall. The Braves have put together a 6-1 surge after losing eight in a row.
Nine of the game's 10 runs came from homers and Miami's Jose Reyes started the long ball outburst with a solo shot in the fifth.
Bourn's was a three-run shot that gave Atlanta a 21-3 scoring advantage in the series.
Mike Minor (3-4) survived five innings to get the win, giving up a lone run on four hits and five walks. He picked up his first victory since April 19.
The loss went to Mark Buehrle (5-6), who left the game after six innings with the Marlins trailing only 2-1.
Miami left 11 men on base while going 0-for-13 with runners in scoring position.