With the game tied 2-2 entering the ninth, Khris Davis led off the frame with a double. Scooter Gennett then reached on a bunt before Overbay smacked a double into the corner in right to clear the bases. Carlos Gomez followed with a single to center to score Overbay.
Later in the inning, Jean Segura reached on a force out, stole second then scored on Aramis Ramirez's base hit to make it 6-2.
Gomez finished 4-for-5 with a pair of RBI and Jonathan Lucroy went 2-for-3 with a solo homer for the Brewers, who dropped two of three to Atlanta in their opening series.
Marco Estrada allowed two runs -- one earned -- on four hits while fanning six over 5 2/3 innings. Brandon Kintzler (1-0) worked a perfect eighth inning to earn the win.
Will Middlebrooks hit a solo homer for the Red Sox, who received their 2013 World Series rings in a pregame ceremony on Friday.