Brewers starter Jimmy Nelson (9-6) absorbed most of the damage, allowing 10 runs -- nine earned -- on 11 hits and three walks while striking out four over 3 2/3 innings.
Nelson, Milwaukee's best pitcher for the last two months, struggled from the start, serving up a one-out home run to Cozart in the first.
The Brewers took a lead in the second on a two-run home run by Manny Pina but the wheels came off for Nelson and the Brewers in the third as the Reds put six runs on the board.
An error on Jonathan Villar aided the rally but four straight hits -- including a two-run single by Scooter Gennett -- seemingly broke the game open for Cincinnati.
The Reds tacked on three more in the fourth. Billy Hamilton drew a leadoff walk, stole second, and then scored on a Cozart single. Joey Votto and Adam Duvall followed with singles. Votto scored when Gennett grounded into a double play.
Eugino Suarez followed with an RBI single to put the Reds up, 10-2, and Nelson was done.
Cincinnati starter Homer Bailey (4-6) gave way after five innings, having allowed a pair of runs on five hits and five walks with five strikeouts. Blake Wood took over to start the sixth and gave up a leadoff single to pinch-hitter Eric Sogard, another to Orlando Arcia and one more to Hernan Perez, which drove in Sogard to make it a 10-3 game.
Villar followed with yet another base hit putting two on with nobody out for Eric Thames, who blasted a 3-2 slider to center for his 26th home run of the season -- and ninth of the year against the Reds.
The Reds turned to Michael Lorenzen but he didn't fare much better.
Ryan Braun reached on a base hit and Travis Shaw walked before Lorenzen struck out Domingo Santana for the inning's first out. Manny Pina reached on a base hit, loading the bases. Milwaukee made it a 10-8 game when Braun came home on a wild pitch.
The Brewers got one more when Sogard bounced into a double play before Lorenzen struck out Arcia to finally end the inning.
Tucker Barnhardt provided some breathing room for the Reds with a leadoff homer in the seventh.
Milwaukee challenged once more, loading the bases with two out in the ninth but couldn't cash in as Raisel Iglesias struck out Jesus Aguilar.
Villar capped off a three-hit day with a solo home run in the ninth, but Iglesias left the tying run stranded when he struck out Travis Shaw to end it.
Nelson, Milwaukee's best pitcher for the last two months, struggled from the start as he served up a one-out home to Cozart in the first.
NOTES: Brewers C Stephen Vogt (left knee strain) began a minor league rehab assignment Friday night with Class A Wisconsin. ... Cincinnati manager Bryan Price RHP Scott Feldman will return from the disabled list and start Saturday against Milwaukee. Feldman has been sidelined with right knee inflammation since July 18. ... With Feldman returning to the rotation, RHP Sal Romano will start the series finale Sunday while RHP Tim Adleman will move into a long relief role out of the bullpen.