The Braves completed a four-game sweep of the Brewers and have won five in a row overall.
Atlanta starter Brandon Beachy allowed one unearned run on four hits and struck out nine in six innings, but did not figure in the decision.
The win went to Jonny Venters (2-0), who worked the top of the eighth before Prado gave the Braves the lead in the bottom half of the inning with his third homer of the year.
Craig Kimbrel produced the 14th and 15th strikeouts by Atlanta pitching in the ninth to earn his seventh save.
Hinske hit his home run in the second and Milwaukee tied it in the fifth on a sacrifice fly from Jonathan Lucroy.
Prado's game-winning homer came off Kameron Loe (2-3), who worked only the eighth inning.