With the Brewers down by a run entering the their final at-bat, Corey Hart singled off Dodgers closer Javy Guerra (1-1). Pinch-runner Carlos Gomez stole second and Mat Gamel drew a walk before Jonathan Lucroy struck out.
Kottaras, pinch-hitting for Cesar Izturis, then laced a 2-2 pitch into the right-center field gap. Gomez scored easily and Gamel slid in under a high throw to the plate with the game-winning run.
Gamel went 2-for-3 with a solo homer for Milwaukee, which broke a four-game losing streak.
Jose Veras (1-0) picked up the victory with a scoreless top of the ninth.
Andre Ethier belted a solo homer for the Dodgers, who sustained only their second loss of the season.