Kershaw (14-2) allowed a run and six hits with two walks and six strikeouts in eight innings. He allowed leadoff hitters to reach base in four of the first five innings of an outing that might qualify as a subpar effort for the two-time Cy Young Award winner.
Nevertheless, the left-handed ace lowered his league-leading ERA to 1.78. Adrian Gonzalez drove in two runs, and scored the go-ahead run in the fifth off Jimmy Nelson (2-3).
Kershaw also reached base three times and had an RBI single in the eighth.
On the mound, Kershaw benefited from base-running blunders and failed aggressive play by the Brewers.
In the third, Nelson wasted his leadoff double after getting tagged out trying to advance from second on a grounder by Carlos Gomez. Later, Gomez was picked off first.
Rickie Weeks hit a leadoff double in fifth. Two batters later, Jean Segura popped up a squeeze attempt caught by a charging Kershaw in front of the plate. He threw easily to third to double off Weeks.
The Brewers staked Nelson to an early 1-0 lead following Ryan Braun's RBI single in the first.
But after a strong first couple innings, Nelson appeared to have trouble keeping his sinker down. He allowed eight hits and two runs in six innings.
Gonzalez increased his season RBI total to 79. His run-scoring double in the seventh came after Carl Crawford singled and move to third following two throwing errors on a pickoff attempt.
DODGERS: Shortstop Hanley Ramirez was placed on the 15-day disabled list with a strained right rib cage muscle retroactive to Saturday. Manager Don Mattingly said the team was being overly cautious, and that Ramirez may need just the minimum time to recover.
BREWERS: Matt Garza, placed on the disabled list on Aug. 5 for a left rib cage muscle strain, won't start throwing again until he is "completely pain free," manager Ron Roenicke said. He added that Garza will likely need more than 15 days to return.
DODGERS: Dan Haren (9-9) is the scheduled starter Monday in Atlanta, though Mattingly said newly-acquired right-hander Kevin Correia (5-13) would likely take a spot turn to give the rest of the rotation extra rest. Los Angeles is in the midst of a 20-day stretch without a day off.
BREWERS: Milwaukee assured itself of a winning homestand against top NL West teams San Francisco and Los Angeles after taking the first two games against the Dodgers. Roenicke promised the team's focus wouldn't change when a four-game series starts Monday against the last-place Cubs in Chicago. Yovani Gallardo (6-6) gets the start.