Matt Szczur also homered off Kershaw (5-5), who fell back to .500 after giving up three runs over seven innings of work.
The Cubs used five relievers after starter Tsuyoshi Wada exited with cramping in his left deltoid muscle. Wada was pulled off giving up a solo home run to Enrique Hernandez and a single to A.J. Ellis to start the third.
While home runs lit up the scoreboard, roughly half of the actual lights at the stadium went out during the bottom of the sixth inning.
After Anthony Rizzo popped out, play was stopped for about 10 minutes due to the poor lighting. Cubs manager Joe Maddon argued home plate umpire Jordan Baker's decision to start play before the lights turned back on and issued a formal protest.
The histrionics were unnecessary when Szczur, promoted from Triple-A Iowa prior to the game, pushed Chicago's lead to 3-1 with a leadoff homer in the seventh. Bryant added another insurance run when he capped his first multi- home run game with a no-doubter off Adam Liberatore in the eighth.
With rain coming down hard, the Dodgers brought the tying run to the plate in the ninth. Joc Pederson hit a solo blast, and Adrian Gonzalez singled with two away, but Motte retired Howie Kendrick on a harmless groundout to secure his second save.
Bryant countered Hernandez's homer in the third by taking a hanging curveball from Kershaw and sending it over the wall in right-center. The star rookie has 10 home runs on the season.
Cubs outfielder Dexter Fowler left the game early due to a sprained left ankle ... Kershaw has given up 11 home runs this season. He allowed 11 in 2013 and nine a year ago ... Kershaw fell one strikeout shy of matching Sandy Koufax and Hideo Nomo as the only Los Angeles Dodgers pitchers with four consecutive starts of double-digit strikeouts ... Dodgers third baseman Justin Turner went 0-for-4, ending a string of 22 consecutive games of reaching base safely as a starter ... Notre Dame head football coach Brian Kelly threw the ceremonial first pitch.