The Brewers halted a four-game losing streak and handed the Pirates their 49th road setback of the season.
Pittsburgh had a 2-1 lead going into the seventh, one of its runs coming in the fifth on Chris Snyder's 12th homer, but the Brewers sent 11 men to the plate to break open the contest.
Prince Fielder walked to open the inning against James McDonald (2-4) and moved to second on a one-out single by Chris Dickerson. Both runners scored on a triple by Escobar.
Escobar came in on a base hit by Jonathan Lucroy, who was singled to second by Joe Inglett.
Corey Hart's double brought home Lucroy and Braun's double delivered both Inglett and Hart to finish the scoring.
The big inning made a winner out of Chris Narveson (10-7), who allowed two runs on seven hits with eight strikeouts over seven innings.
Swim Week Miami Beach 2014 [PHOTOS]