The Dodgers trailed twice in their first meeting with the Braves since eliminating them in a National League division series last season.
Puig came up a home run short of hitting for the cycle, lifting his team-leading average to .317 despite getting hit on the head by the ball while sliding head-first in the seventh. He struck out in his final at-bat.
Los Angeles rallied from a two-run deficit to tie it 4-all on an RBI double by Adrian Gonzalez and a RBI single by Carl Crawford in the fifth.
That set the stage for outfielders Crawford and Kemp to put the Dodgers ahead in the seventh.
