Polish pair Robert Lewandowski and Jakub Blaszczykowski shared the other goals in another convincing performance by Juergen Klopp's team.
Gotze set Dortmund on their way in the third minute, scoring from close range after Kevin Grosskreutz headed across the goal, and Blaszczykowksi scored the second with a deflected shot in the 18th minute.
Dortmund made it 3-0 before half-time with a Goetze free-kick which went under the Hannover wall.
Mame Biram Diouf pulled one back before Gotze and Lewandowski added further goals in the last 10 minutes.
Christian Trasch and Bas Dost scored in the last 15 minutes as VfL Wolfsburg came from behind to beat Bayer Leverkusen 2-1 after the visitors, second in the Bundesliga, had gone ahead in the first half thanks to a Fagner own goal.
First-half goals from Christian Gentner and Vedad Ibisevic set VfB Stuttgart on the way to a 2-1 win over second division Cologne, who pulled one back in the second half through Christian Clemens.
VfL Bochum brushed aside TSV Munich in an all-second division tie.
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