Zack Cozart and Todd Frazier each stole twice, and Kristopher Negron added one more.
The Reds stole six times in the first four innings, using six hits and four walks to build a 6-0 lead against Jake Arrieta (7-5).
Cincinnati stole three in a three-run second inning and two in a three-run fourth.
Dylan Axelrod (1-0) pitched five scoreless innings of two-hit ball and struck out eight.
Cubs rookie Jorge Soler doubled and singled in his second major league game and drove in a run. He homered and singled in his debut on Wednesday.