Frazier's home run, his fourth of the year, came on an 0-1 pitch off Cristhian Martinez (2-1) and just cleared the wall in right. Cincinnati won its fifth in a row, the Reds' longest such streak of the season.
The score had been tied since the sixth, when Cincinnati's Jay Bruce drove in Zach Cozart from third on a fielder's choice grounder. Dan Uggla produced Atlanta's run with a homer in the fourth.
Cincinnati starter Bronson Arroyo made the 300th start of his major league career but did not figure in the decision. Aroldis Chapman (4-0) struck out two during a perfect top of the ninth to get the win.
Chapman has thrown 24 1/3 innings of relief this season and has not allowed an earned run.