Bryant recorded the first multi-homer game of his career with his ninth and 10th blasts of the season to help the Cubs to their third straight win.
After starter Tsuyoshi Wadaexited in the third inning with a left shoulder cramp, five different relievers combined to allow one run in seven innings of work.
Reliever Travis Wood came in for Wada in the third and pitched 3 1/3 scoreless innings. James Russell, Pedro Strop and Hector Rondon followed without allowing a run before Jason Motte allowed a run on Joc Pederson's home run in the ninth.
"I definitely liked Woody in that game today against that lineup," said Cubs manager Joe Maddon. "And I thought Rondon threw the ball as good as I've seen him in a while."
The bullpen outlasted Kershaw, who gave up three Chicago runs. He went seven innings and allowed four hits while striking out nine.
"I thought he was pretty good, honestly," manager Don Mattingly said of Kershaw. "He left one breaking ball up to Bryant, and the ball to Szczur got back to the middle of the plate. Other than that, pretty good."
Kershaw didn't receive much offensive support, though, as Kiké Hernandez's home run in the third and Pederson's in the ninth -- his 19th long ball of the season -- were two of five hits for the Dodgers in the game.
MOMENTS THAT MATTERED Bryant's blasts: After he had gone without a hit in his previous 15 at-bats, Bryant delivered as the Cubs trailed in the third inning, taking Kershaw's 1-2 pitch to right field, also scoring Dexter Fowler. Bryant homered again in the eighth inning, a solo shot to left field off Adam Liberatore.
"They forced him to go out for the curtain call, but he comes back in and gives you a high five like, 'I'm going to do this again,'" Maddon said. More >
Pump-faked: With Pederson on first base and Yasiel Puig on second base after two-out walks in the third inning, and hot hitter Justin Turner up, Puig was picked off second base by Cubs catcher and former Dodger David Ross, who pump-faked Puig before throwing down to second base before Puig could return.
Kershaw a year later: The home runs by Bryant and Szczur were the 10th and 11th allowed by Kershaw this year, two more than all of 2014.
"He was just really deceptive," Bryant said. "But he's the best pitcher in the game, and I just got a good pitch to hit." More >
Saved by the double play: Though it wasn't anything flashy, the Cubs' bullpen picked up some timely help from its defense to escape jams. In the third inning, with a runner on first, Kershaw grounded out to reliever -- and winning pitcher -- Wood, who fired to Starlin Castro at second base, who fired to Addison Russell covering at first. In the fifth, Ross picked off Hernandez following a strikeout to end the inning. And in the sixth, reliever James Russell got Pederson to hit into a 4-6-3 double play to end that inning.
PROTEST OF WILLS The Cubs led, 2-1, when some lights went out at Wrigley Field in the sixth inning, and Maddon became concerned about facing Kershaw with a disadvantage.
"'Be a little more patient and wait for the lights,'" said Maddon. "That was my argument, that's it. Nothing more complicated than that. I just thought it was inappropriate, and I made my case."
Umpires originally told Maddon he could not protest what they deemed a judgment call. However, Maddon argued that judgment calls did not include the lights and was allowed to protest.
"I just wanted to say, 'Get off the field and keep the game going,'" Kershaw said. "But standing around for 10 minutes, my legs were getting heavy. I wanted an answer. I don't know if Joe was trying to do it on purpose or what? It didn't affect me, but it was a good idea." More >
QUOTABLE "I just have so much faith in him. When a guy goes through a drop like that, it just takes one night." -- Maddon on Bryant breaking out of his slump for a career night.
WHAT'S NEXT Dodgers:Zack Greinke has a 1.77 ERA and has allowed only one home run in 35 2/3 innings in road starts this season. He is coming into Tuesday's 5:05 p.m. PT start off seven scoreless innings on four hits against Texas, but he hasn't won since May 5.
Cubs:Jason Hammel looks to get back on track as he takes the mound for the Cubs. In his previous two outings, Hammel has allowed a combined seven runs (four earned) and 10 hits, both of which were no-decisions. First pitch is scheduled for 7:05 p.m. CT.