The score had been tied since the eighth inning, when the Braves scored twice to keep alive their chances for what turned out to be a fourth straight win.
Ross began the two-out rally in the 11th with a single off Brandon League (0-6) and moved to third on a base hit by Paul Janish. Francisco came on to pinch-hit for pitcher Jonny Venters and delivered the winning blow on a 1-1 pitch.
The Dodgers, who went into the evening leading San Francisco by a half game at the top of the National League West, dropped their second in a row.
Venters (5-3) got the win, working the top of the 11th, during which he benefited from a double play to escape trouble.