The Dodgers were 14-5 winners over the Chicago Cubs, helping the Dodgers' record improve to 88-66 and see them close on the National League's West division title.
Mathematically, though, they are assured of at least a wildcard after the victory, which saw the Dodgers put on two six-run innings.
AJ Ellis (two hits, four RBIs) and Yasiel Puig (two hits, three RBIs) were the stars with the bat, while Kershaw's record moved to 20-3 after he struck out nine in five innings work.
The San Francisco Giants are 3.5 games off the Dodgers, after a 5-0 shutout loss to the San Diego Padres.
In the NL wildcard race, the Pittsburgh Pirates downed closest rivals the Milwaukee Brewers 4-2, scoring all of their runs in the eighth inning.
Russell Martin bludgeoned a three-run shot to power the Pirates' comeback, sending them to a fifth straight win and opening a 4.5-game gap on the Brewers.
The Pirates made no ground in their division chase, though, as NL Central leaders the St Louis Cardinals were 2-1 winners over the Cincinnati Reds.
The Cardinals hold a 2.5-game lead in the division.
In the only live remaining American League division, the Detroit Tigers won a crucial clash with the Kansas City Royals 10-1 to move 1.5 games clear.
A strong performance from pitcher Justin Verlander plus three RBIs from Ian Kinsler sent the Tigers to the convincing win, fending off their only challenger for the title.
The Royals remain in the second AL wildcard spot, but only just, with the Seattle Mariners' 10-5 win over the Houston Astros taking them within half-a-game of the post-season.
The Oakland Athletics hold the other wildcard, after a much-needed 3-1 win over the Philadelphia Phillies.
Michael Cuddyer broke out with seven RBIs as the Colorado Rockies trounced the Arizona Diamondbacks 15-3, the Minnesota Twins were 5-4 walk-off winners over the Cleveland Indians, the New York Mets shut out the Atlanta Braves 5-0 and the Washington Nationals edged the Miami Marlins 3-2.
The Chicago White Sox edged the Tampa Bay Rays 4-3, the New York Yankees inflicted a sixth straight loss on the Toronto Blue Jays 5-3, the Boston Red Sox downed the Baltimore Orioles 5-3 after 10 innings and the Texas Rangers defeated the Los Angeles Angels 12-3.