Chicago had a 3-1 advantage going into the eighth, but Esmailin Caridad (0-1) struggled while throwing 30 pitches in the inning and allowing all four runs.
That brought up Stubbs, who hit a 1-0 pitch over the wall in center.
Micah Owings (1-0) threw three scoreless relief innings to get the victory. Cincinnati had lost two out of three to open the season against St. Louis.
Derrek Lee had three hits, including a solo homer, and drove in two runs for Chicago. The Cubs have lost three out of four.
Notable deaths of 2014 [PHOTOS]