The Braves boosted their advantage in the National League East to nine games over both Washington and Philadelphia, by far the most one-sided division lead in baseball.
Minor (10-5) did not walk a batter and struck out five. The only St. Louis run came on a second-inning homer by Yadier Molina.
Craig Kimbrel struck out two in the ninth while recording his 29th save.
Minor helped his own cause with an RBI single during Atlanta's two-run second. The Braves padded their lead when Jason Heyward hit a solo homer in the fifth and Atlanta added an unearned run in the seventh on a throwing error by St. Louis shortstop Pete Kozma.
Adam Wainwright (13-6) suffered the loss, giving up all four runs on seven hits in seven innings. He struck out eight while trying to add to the largest win total in the National League.