Chipper Jones cracked a solo home run for the Braves to contribute to the win and also mark his 40th birthday.
Tyler Pastornicky scored the game-winner on Prado's triple and collected three hits in the game.
The Braves snapped a two-game losing streak and handed the Dodgers' their first loss in eight home games.
Juan Rivera drove in two runs with a first-inning homer for Los Angeles and was one of four Dodgers to have two hits in the game.
Javy Guerra (1-2) took the loss in relief.
Both starters -- Aaron Harang of the Dodgers and Atlanta's Mike Minor -- lasted six innings and departed with the score tied 3-3.