It took three minutes following his introduction for the 20-year-old midfielder to score in the 81st minute, after Czech defender Theodor Gebre Selassie equalized for Bremen on his league debut in the 75th.
''I'm delighted with how it worked out,'' said Goetze, who missed most of the second half of last season through injury. ''For the team and myself.''
Marco Reus had opened the scoring in the 11th minute to repay some of the ?17.5 million ($21.9 million) Dortmund paid Borussia Moenchengladbach for the German player of the year in the offseason.
The 23-year-old striker could not have made a better start on his hometown return. Jakub Blaszczykowski sent the ball out to the right, and Reus capitalized on Aleksandar Ignjovski's indecision to overtake the midfielder and tuck the ball inside the far post.
Reus said he ''had goosebumps walking out onto the pitch'' before the game.
Despite Dortmund's growing influence, Eljero Elia would have equalized but for a good block from Roman Weidenfeller. Then Marco Arnautovic hit the post.
''We made life difficult for ourselves,'' Reus said.
Sebastian Mielitz pulled off a great save to deny Mats Hummels' header early in the second half, but Elia and the on-loan Chelsea midfielder Kevin de Bruyne continued to threaten.
Bremen's equalizer was deserved when it came. Arnautovic floated in a cross and offseason signing Selassie rose between two Dortmund defenders to send a glancing header over the helpless Weidenfeller.
Dortmund coach Juergen Klopp reacted by introducing Goetze, and the switch had the desired effect when the midfielder controlled Robert Lewandowski's through ball with his left foot and then swept it past Mielitz with his right.
''The coach told me to give it socks,'' Goetze said. ''I did that, I think.''
Dortmund stretched its unbeaten league run to 29 games 49 years to the day after the first Bundesliga games were played.
Last season's runner-up, Bayern Munich, visits promoted Greuther Fuerth on Saturday.