Kershaw set the tone with a season-high 14 strikeouts to reach 2,000 for his career. He held the Brewers to two runs and a walk in seven innings, but Brewers starter Jimmy Nelson matched him pitch for pitch and set his own season high by fanning 11 in eight innings.
"He pitched better than I did," said Kershaw, who reached 2,000 strikeouts in 1,837 2/3 career innings, which is the third-fastest in major league history behind Pedro Martinez (1,715 1/3) and Randy Johnson (1,734), according to the Dodgers.
After allowing his first two batters of the game to reach, Kershaw retired 20 batters in a row a before Domingo Santana's two-out solo homer gave Milwaukee a 1-0 lead in the seventh.
Kershaw appeared to be on track to take a loss for the first time in five career starts at Miller Park because his offense could do nothing against Nelson, who held Los Angeles scoreless on five hits and didn't walk a batter for the second consecutive start.
"Jimmy did his job," Brewers manager Craig Counsell said. "He pitched brilliantly. It was a heck of a pitcher's duel."
Even with the slimmest of leads, Milwaukee felt good heading into the ninth. Right-hander Corey Knebel had been dominant all season and had converted four of five save opportunities while holding opponents to a .111 batting average since Neftali Feliz was removed from the closer's role May 12.
Knebel got off to a good start, striking out Corey Seager to open the ninth, but he missed his mark on a 3-1 fastball to Yasmani Grandal, who dropped it in the right field stands for his sixth homer, tying the score at 1.
"He just fell behind to Grandal and he put a good swing on it," Counsell said.
After Jacob Barnes and Carlos Torres preserved the tie in the 10th and 11th innings, Counsell turned to Feliz for the 12th, but he gave up a leadoff homer to Bellinger that put the Dodgers ahead for good.
"I thought Neftali made a decent pitch to Bellinger," Counsell said. "I did not think it was a bad pitch. He executed a pitch that was up. I give him a lot of credit right there. It was an 'up' pitch, which is what we wanted to do. He got on top of a 95 mph fastball right there.
"The high fastball gave a bunch of hitters a lot of trouble tonight. I give the guy credit for hitting it."
Along with Milwaukee's 26 strikeouts -- the most ever by a Dodgers opponent -- the Brewers' pitching staff had 16 more. The combined 42 punchouts set a National League record for most combined strikeouts in a game with 42.
"I've never seen anything like that," Roberts said. "But for our guys to punch out 26 guys, and 42 overall ... We saw Nelson last year. He's got really good stuff. He's kind of changed his mix going with the four-seamer a little bit more. His stuff is electric. In the zone, there's life in the zone. He really matched Clayton pitch by pitch.
"As a baseball fan, it was really fun to watch."
NOTES: Brewers OF Ryan Braun ran on an anti-gravity treadmill Monday for the first time since returning to the disabled list, but there is still no timetable for his return, manager Craig Counsell said. Braun played just six games last month because of his strained left calf. ... Dodgers pitchers struck out 26 batters, setting a franchise record. ... Milwaukee RHP Jimmy Nelson became the first pitcher in franchise history with 10 or more strikeouts without walking a batter in consecutive starts. ... Dodgers LHP Clayton Kershaw surpassed 2,000 career strikeouts by fanning Brewers 2B Jonathan Villar in the fourth inning. He accomplished the feat in 277 games. Only Randy Johnson, who needed 262 games, did it in fewer games in major league history. ... The 42 combined strikeouts broke the previous NL record (40) set on June 19, 2001, by the Giants and Padres, who struck out 20 times each in a 15-inning game.