Starting pitcher Clayton Kershaw kept Atlanta's batters quiet after initially struggling and pitched seven innings, allowing just one run on three hits to set up the win.
The Braves managed just five hits and struck out 15 times, looking a pale shadow of the team who had an impressive 56-25 home record during the regular season.
The Dodgers jumped on the Braves early, charging to a 4-0 in the top of the third inning when Adrian Gonzales slugged a two-out, two-run homer.
Atlanta's starting pitcher Kris Medlen lasted just four innings in his second post-season game, giving up nine hits and five runs.
It was a total team effort by the Dodgers, as seven players managed at least one hit to give Don Mattingly his first post-season victory as a major leagues manager.
Game Two will be in Atlanta on Friday.