The game's only run scored after Colorado starter Jhoulys Chacin missed a throw to the mound after a pitch.
Atlanta also took Wednesday's game 1-0 and Thursday's result marked the first time the Braves won -- and the first time the Rockies lost -- consecutive 1-0 games. It was from far from pretty, however.
Atlanta pitchers only got the Rockies out in order twice in the game. Hudson (14-5), for instance, pitched out of a jam when the Rockies opened the third inning with back-to-back singles.
Hudson allowed six hits over seven innings. He struck out two and walked two. Peter Moylan and Eric O'Flaherty got through the eighth 1-2-3 before Craig Kimbrel pitched a one-hit ninth for his 34th save.
Chacin (2-5) gave up seven hits and three walks over 3 1/3 innings but the only run in the game was unearned.
Juan Francisco opened the Atlanta second with a bunt single and took third on a ground-rule double from Dan Uggla. Three pitches into the next batter, Chacin missed an easy toss back from catcher Wilin Rosario and Francisco raced home for what proved to be the only run of the game.
Notable deaths of 2014 [PHOTOS]
Michael Sam cut from the Dallas Cowboys