The Brewers were trailing 3-1 in the fifth when they erupted for five runs to grab control of the contest, triggered when Juan Francisco drew a lead-off walk from Cubs starter Edwin Jackson and Weeks drove him in with a double.
After going to third on a wild pitch, Weeks scored the tying run on Caleb Gindl's one-out single and Jean Segura followed with an RBI double, putting Milwaukee ahead to stay.
Weeks added a two-run homer in the sixth and a solo shot in the eighth, finishing 3-for-4 with four RBI.
Jonathan Lucroy also drove in a pair of runs for Milwaukee, which took the opener of a three-game series.
Kyle Lohse (3-6) recorded the victory, yielding three runs on eight hits over seven innings. He didn't issue a walk and struck out three.
Jackson (3-10) was tagged for six runs on six hits in a 4 2/3-inning performance.
Nate Schierholtz belted a three-run homer for the Cubs.