Pedro Strop closed out a tense ninth inning as Chicago boosted its NL Central lead to four games over the second-place Brewers.
Quintana (12-9) allowed two runs in 6 2/3 innings, improving to 3-1 at Miller Park with the Cubs.
Strop allowed a walk and a single to open the ninth but got the final three outs, striking out Curtis Granderson with two runners on to preserve the win.
Granderson belted his first home run as a member of the Brewers in a pinch-hit role in the seventh, a two-run shot that cut Chicago's lead to 6-4.
The Cubs finally broke their string of 16 scoreless innings against Milwaukee right-hander Jhoulys Chacin (14-6) when Murphy led off the fourth with an opposite-field homer into the Brewers' left-field bullpen.
Javier Baez followed with a single and his aggressive baserunning helped the Cubs extend their lead. He raced for third on a single by Anthony Rizzo, and when center fielder Lorenzo Cain's throw sailed, Baez scored and Rizzo went to third on a second error by shortstop Orlando Arcia. Ben Zobrist doubled inside the right-field line to score Rizzo and give the Cubs a 3-0 lead.
A soft RBI single by Willson Contreras capped the four-run rally.
Jonathan Scoop's bad-hop single drove home a run in the bottom of the inning, and the Brewers loaded the bases with two outs. Quintana induced Arcia to hit a pop fly to shortstop to escape the jam.
Zobrist drove in his second run in the fifth with a two-out single, but Rizzo was thrown out at the plate trying to score from second base.
Schwarber homered into the second deck in right field in the sixth to make it 6-1.
Quintana allowed a leadoff single to Jesus Aguilar and a double to Ryan Braun in the sixth, but the Brewers managed just one run on a groundout by Schoop.