Hand, who had made eight relief appearances for Milwaukee before being called on to make a spot start, yielded just two hits and a walk and struck out three before exiting.
Burke Badenhop (1-3) followed up with 1 1/3 shutout frames to earn the decision. Michael Gonzalez and John Axford blanked the Braves for one inning each.
Francisco Rodriguez finished up with his 300th career save and sixth of the season as the Brewers whitewashed Atlanta for a second straight contest.
Tim Hudson (4-7) was saddled with loss after yielding two runs on seven hits over six frames for National League East-leading Atlanta, which dropped its eighth straight at Miller Park.