Cozart's RBI single ignited a four-run second inning and his solo homer in the fifth gave Cincinnati an 8-2 lead.
Devin Mesoraco added three hits, including a three-run shot, while Todd Frazier added two RBI and Joey Votto belted a bases-empty homer as the Reds banged out 15 hits in the rout.
Coupled with their 11-0 victory in Monday's four-game series opener, the Reds outscored the Giants 20-3 over the first two contests.
Tony Cingrani (4-1) went 6 2/3 innings to earn the decision, allowing two runs on five hits. He walked three and struck out two.
Giants starter Eric Surkamp (0-1), called up from Class AAA Fresno before the game, was pounded for seven runs on nine hits over just 2 2/3 frames.