Weeks, who came into the game with only seven hits in 72 at-bats, singled and doubled before connecting for his three-run blast in the eighth inning as the Brewers won for the 12th time in 15 games.
Jean Segura and Yuniesky Betancourt also went deep as Milwaukee pounded out 15 hits en route to clinching their three-game home series.
Ryan Braun went 3-for-5 with two runs scored in the triumph.
Tom Gorzelanny (1-0) earned the decision with a scoreless seventh inning after starter Marco Estrada had yielded five runs on seven hits over five full frames.
Pittsburgh's Bryan Morris (0-1) absorbed the loss.
Andrew McCutchen went 4-for-5 with a solo homer and two RBI while Starling Marte notched three hits, including a three-run shot, for the Pirates.