The Dodgers added a run in the sixth when Mark Ellis, who singled, scored on Hanley Ramirez's double off Jordan Walden.
The Braves' only run came in the fourth when Freddie Freeman, who hit a one-out single, scored on Chris Johnson's single. Johnson's hit came after Brian McCann's long fly ball was caught at the left-field wall by Crawford.
"I thought I got enough to get it out," McCann said. "But I didn't."
Game 2 in the best-of-five series is Friday night in Atlanta, with Zack Greinke (15-4) starting for the Dodgers against Mike Minor (13-9).
NOTES: OF Andre Ethier hit a groundout when he pinch hit for Kershaw in the eighth. Manager Don Mattingly said he plans to use Ethier, who has a left ankle injury and hasn't played in the field since Sept. 13, only as a pinch-hitter in the series. ... Braves 2B Dan Uggla said he was "disappointed and upset" to be left off the Atlanta roster. ... The attendance was 43,021. There were sections of empty seats in the upper level in left field.