The Brewers scored a run in the eighth, ninth and 10th innings to overcome a 4-2 deficit and end a five-game losing streak.
Prince Fielder brought Milwaukee within one in the eighth with his 19th home run and the Brewers tied it in the ninth on a one-out triple by George Kottaras that brought home Carlos Gomez from third.
With Evan Meek (4-3) on the mound in the 10th, Jim Edmonds singled to open the inning and was erased on a fielder's choice grounder from Weeks.
Weeks then stole second and, after Corey Hart struck out, Fielder was walked intentionally to bring up Braun. He drove a pitch over the head of right fielder Ryan Church for the winning hit.
John Axford (4-1) got the win by working the top of the 10th, during which he stranded a runner on third by striking out both Pedro Alvarez and Ryan Doumit.
Pittsburgh's road record for the season fell to 11-36.
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