Jones' homer came in the third inning and put the Braves ahead for good at 4-1.
Jason Heyward added a solo shot and Eric Hinkse contributed a two-run single for the Braves, who completed a three-game sweep of the Brewers and notched their fifth straight win.
Beachy (1-1) yielded only one unearned run on three hits over seven frames, walking two and striking out six.
Milwaukee starter Chris Narveson (1-1) was tagged for five runs on four hits before being yanked after four-plus innings.
Jonathan Lucroy went 2-for-4 with a pair of home runs for the Brewers, who dropped their fourth in a row and eighth straight in Atlanta.