Leading 4-3 on aggregate following an enthralling first leg in Norway last week, Thomas Tuchel's side were on top from the first whistle but they were denied an early breakthrough when Sondre Rossbach managed to keep out two efforts from Matthias Ginter and deny Jonas Hofmann.
It was the visitors who broke the deadlock when Ole Jorgen Halvorsen took advantage of poor defending to powerfully head home from a corner.
The Dortmund response was swift and emphatic as Henrikh Mkhitaryan produced a moment of brilliance to turn into space before picking out the top corner with a stunning strike.
Any hopes of an unlikely fightback from Odd then appeared to be dashed when Reus fired home his second after Gundogan and Mkhitaryan had combined to set up the opportunity for their teammate.
However, there were more celebrations to come from the home supporters before the interval as Ginter's low cross evaded a couple of Odd defenders to give Shinji Kagawa the simple task of slotting home to make it 8-4 on aggregate.