Cozart delivered a three-run homer and an RBI single, while Negron ended 3- for-4 with a two-run homer and three runs scored in the Reds' seventh straight victory over their in-state rivals in Cincinnati. Jeff Frazier added a pair of hits, including a two-RBI double.
Mat Latos (4-3) aided the Reds' cause by taking a shutout into the eighth inning before being reached for three runs over his 7 2/3-frame stint.
Danny Salazar (4-5) struck out five in four innings for Cleveland, but served up both Cozart's and Negron's homers for a total of five runs allowed.
Jason Kipnis had three of the Indians' seven hits, with Yan Gomes finishing 2- for-4 with a solo homer and Ryan Raburn contributing a two-run double.