The Red Sox took advantage of two more errors to score twice in the eighth to tie it 3-all and set up closer Koji Uehara (1-1) for the win.
Bradley and Brock Holt drew back-to-back walks off closer Craig Kimbrel (0-1). Xander Bogaerts followed with a hard grounder to Chris Johnson at third base for an infield single. Johnson stopped it and made a strong throw but La Stella missed the catch with Holt bearing down. Bradley scored and the Red Sox rushed out of the dugout as the ball trickled into right field.
Boston completed a four-game sweep in the home-and-home series with the Braves.
Jason Heyward homered and Freddie Freeman drove in a run and scored once for the Braves, who lost their fourth straight and fifth in six games.
Atlanta led 3-1 when Holt led off the eighth with a bloop single to left off reliever David Carpenter. He went to second when left fielder Justin Upton took his eye off the ball and it bounced past him. Bogaerts followed with an RBI single and Dustin Pedroia reached on a slow roller up the middle that was called a hit.
A.J. Pierzynski followed with another single to center that B.J. Upton let get by him for the second error of the inning, allowing Bogaerts to score and tie it at 3-all.
Atlanta had just gone up 3-1 with a run in the top of the eighth. B.J. Upton led off with a single to left and scored when Freeman followed with a double off the wall in left-center.
Heyward’s homer with two outs in the third gave Atlanta a 1-0 lead. The Braves added a run in the fourth when Freeman led off with a walk, advanced to second on a balk and scored on Evan Gattis’ single.
Boston cut the lead to 2-1 with a run in the fifth when David Ross doubled and scored on a two-out double by Brock Holt.
Boston starter Jake Peavy pitched eight innings, allowing three runs on eight hits. He struck out four and walked one.
Mike Minor also had a strong start for Atlanta, holding Boston to one run on seven hits in seven innings. Minor didn’t walk a batter and struck out three.
NOTES: Boston returns to American League play tonight when Tampa Bay visits, with LHP David Price (4-4) facing Boston righty Brandon Workman (0-0). ... Atlanta will travel to NL East co-leader Miami. The Braves’ Julio Teheran (4-3) will face Tom Koehler (4-4) . ... Boston DH David Ortiz did not start because of a sore right calf, a move manager John Farrell called precautionary. Ortiz pinch-hit in the eighth and drew an intentional walk with two outs. ... Boston 1B Ryan Lavarnway left the game in the sixth with a sore left wrist. ... ... Atlanta SS Andrelton Simmons returned to the lineup after missing Wednesday’s game with an inflamed right ankle.