Uggla delivered a two-run double and later scored along with Brian McCann when Francisco supplied a pinch-hit single in the decisive frame, handing the Braves a 6-1 lead.
Tommy Hanson (12-5) earned the victory, holding the Marlins to one run on six hits while walking three over five innings. The right-hander struck out four.
Marlins starter Mark Buehrle (9-10) was tagged for six runs on eight hits, lasting just 4 1/3 frames.
Jose Reyes got Miami's only RBI.