Jason Heyward also homered for the Braves, who have won five in a row after losing their first four games. Beachy gave up three hits and one unearned run in seven innings.
Jones, whose 40th birthday is April 24, had arthroscopic surgery on March 26 to repair a torn meniscus in his left knee. He missed the first two games of the series after having the knee drained on Friday.
Chris Narveson gave up five runs in four innings for the Brewers, who have lost four in a row. Jonathan Lucroy homered and drove in three runs.
Jones has two homers in only three games. After missing the first four games of the season while recovering from the surgery, he homered in his first game back on Tuesday night in Houston. He announced in spring training he will retire after the season.
Beachy (1-1) became Atlanta's second straight starter to pitch at least seven innings without allowing an earned run. He struck out six and walked two batters. Mike Minor gave up two hits and one unearned run in 7 1-3 innings in the Braves' 2-1 win on Saturday night.
Milwaukee's first run came after Braves left fielder Martin Prado, fighting the sun, dropped Aramis Ramirez's flyball to open the second inning. Ramirez advanced to second on the play and scored on Lucroy's two-out single to center.
Beachy gained momentum by striking out Ramirez and Hart with runners on first and second in the third inning.
The Braves scored one run in the first off Narveson (1-1). Michael Bourn doubled, moved to third on Prado's sacrifice and scored on Brian McCann's line drive to right field.
Jones' three-run homer in the third drove in Beachy, who walked, and Prado, who doubled to left. Heyward added his homer high and deep in the right-field seats in the fourth.
The Braves loaded the bases in the eighth against Brewers closer John Axford, who had not pitched since Wednesday. After Uggla walked, first baseman Mat Gamel bobbled a grounder by Freddie Freeman for an error. Weeks added a throwing error on the play, making a poor toss toward first.