Hudson (4-2) produced his 13th career shutout, mastering the Marlins with a 108-pitch performance. The right-hander issued three walks and struck out three while not allowing a runner to reach third base until the ninth inning.
Uggla, meanwhile, went 4-for-5 with a solo homer in the fourth and a two-run shot in the ninth. In between, he delivered a two-run single to lead a 16-hit Braves attack.
Andrelton Simmons added three hits and knocked in three runs, while Jason Heyward went 3-for-4 with an RBI for Atlanta, which claimed the first of a three-game set in Miami.
Anibal Sanchez (3-4) was roughed up for seven runs on eight hits over 6 2/3 frames for the Marlins, who came into the game tied for first place in the National League East.