Needing a win to avoid elimination, Gonzalez never got the ball to lights-out closer Craig Kimbrel.
"You don't want it to ever end the way we ended today. But we had the right guy out there. Carp has been good for us," Gonzalez said. "There is nothing to be ashamed of."
The Dodgers gambled in bringing back Kershaw on three days' rest for the first time in his career. But with a chance to close out the series, they opted for their ace over scheduled starter Ricky Nolasco. The move paid off when the 2011 Cy Young Award winner tossed six solid innings before turning it over to the bullpen.
"This is the postseason," Kershaw said. "You just go. It's a one-month sprint and I'm looking forward to the next couple games."
Kershaw wasted a 2-0 lead, giving up two unearned runs and three hits. He struck out six and walked one on 91 pitches.
With one out in the seventh, Elliot Johnson tripled into the right-field corner, sending Puig sliding into the dirt chasing the ball as it caromed off the wall. Constanza, batting for starting pitcher Freddy Garcia, singled into center to put the NL East champions ahead for the first time.
Garcia allowed two runs and eight hits in six innings. He struck out six and walked two.
The Braves tied it 2-all in the fourth.
Crawford's second homer of the game came in the third, landing deep in the lower right-field seats for a 2-0 lead.
Crawford led off the first with a towering home run to the right-field pavilion, smiling broadly as the Dodgers took a 1-0 lead.
"The smile on his face when he hit that homer is priceless," Mattingly said.
NOTES: The Dodgers lost in five games to Philadelphia four years ago in the NLCS. ... The last Dodgers pitcher to start on short rest in the playoffs was Derek Lowe, who allowed two runs and six hits in five innings of a no-decision in Game 4 of the NLCS on Oct. 13, 2008, against Philadelphia. ... Crawford's leadoff homer was the second in Dodgers postseason history. Davey Lopes homered leading off Game 6 of the 1978 World Series. He's now the first base coach for the team. ... Kershaw allowed seven unearned runs during the regular season, all of them at Dodger Stadium.