Trailing 4-1 heading into the frame, the Brewers scored five times to grab control of the contest.
Nyjer Morgan drew a walk from Jenrry Mejia (0-1) to lead off the frame and Jean Segura reached on a bunt single. That brought up pinch-hitter Logan Schafer, who laced a pitch into the right-field corner for a two-run triple.
Norichika Aoki walked, chasing Mejia, and Jeremy Hefner relieved, only to uncork a wild pitch that brought Schafer home with the tying run.
The next batter, Weeks, then slammed a 1-2 pitch deep over the left center field fence for a two-run shot and Milwaukee went on to win for the 10th time in 11 home games.
Weeks finished 2-for-4 with three RBI and reliever Brandon Kintzler (2-0) earned the win for the Brewers, who entered the contest 3 1/2 games out of a National League wild card spot.