Bedard (1-4), who signed a one-year contract with the Pirates during the off-season, struck out nine over five innings, yielding one run on five hits and two walks.
The left-hander had racked up losses in each of his first four Pittsburgh starts.
After Chris Resop gave up the Braves' second run in the sixth, Juan Cruz and Jason Grilli each tossed an inning of scoreless relief and Joel Hanrahan collected his fourth save with a 1-2-3 ninth as the Bucs broke a two-game losing streak.
Braves starter Randall Delgado (2-2) surrendered four runs on eight hits over a 4 1/3-inning effort.
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