Jones -- named to the 2012 All-Star Game earlier Tuesday to replace the injured Matt Kemp of the Los Angeles Dodgers -- delivered a bases-clearing double and added a run-scoring single for the Braves, who erased a 3-1 deficit.
Bourn's fifth-inning triple put Atlanta ahead to stay and Martin Prado added a pair of RBI as Atlanta evened its four-game series with Chicago at one win apiece.
Jair Jurrjens (2-2) worked around nine hits surrendered over five innings to earn the victory, allowing three unearned runs while walking none and striking out two.
Bryan LaHair supplied two hits and an RBI for the Cubs, who saw their four-game winning streak snapped.
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