After scoring twice in the bottom of the eighth to tie the contest at 3-3, Boston capitalized on back-to-back walks issued by Braves closer Craig Kimbrel (0-1) to begin the Red Sox' half of the ninth.
Xander Bogaerts followed with a sharp grounder to third on which Chris Johnson made a diving stop but threw past Tommy La Stella covering second, with Bradley alertly circling third on the miscue to complete the comeback.
The triumph was Boston's fourth in a row following a dreadful 10-game skid. All of them have come at the expense of the Braves, who dropped two to the Sox in Atlanta earlier in the week prior to this two-game set at Fenway.
Bogaerts and Brock Holt each collected three hits with an RBI, with Jake Peavy tossing eight strong innings of three-run ball for Boston in a no-decision. Koji Uehara (1-1) held Atlanta scoreless in the top of the ninth to pick up the win.
The Red Sox managed just one run on seven hits against Mike Minor in the Braves starter's seven innings of work. Jason Heyward paced Atlanta offensively with two hits, including a solo homer.