After coming from three goals behind to win the first leg 4-3, Thomas Tuchel's side handed the visitors an early boost before cruising to an emphatic win to claim an 11-5 aggregate triumph at Signal Iduna Park.
Odd raised hopes of an unlikely fightback following their first-leg collapse when poor marking allowed Ole Jorgen Halvorsen to head home from a corner.
The Norwegian side lost any momentum gained by the early breakthrough as Dortmund hit back quickly through Henrikh Mkhitaryan, who capitalised on space offered to him by picking out the top corner with a strike from the edge of the box.
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Tuchel's side moved further ahead on aggregate just two minutes later when Ikay Gundogan broke into the box before pulling the ball back to allow Reus to fire home.
Gundogan was involved once again for his team's third of the night as he linked up with Mkhitaryan to set up Reus for a simple close-range finish.
Dortmund's first-half fightback ensured that they could relax after the break, and they were gifted a fifth goal when the influential Gundogan unleashed a long-range strike that Rossbach could only parry into his own net.
Chances continued to arrive on a regular basis for the hosts and Odd were powerless to prevent the Dortmund attack from finding space as Reus completed his hat-trick by poking home from Mkhitaryan's inch-perfect cross.
On a miserable evening for Dag-Eilev Fagermo's side, there was a consolation when Oliver Berg finished emphatically after reacting quickest to a loose ball in the box.
It was Dortmund who had the final say, though, as they completed a sixth successive win in all competitions when Kagawa added his second by converting from Pierre-Emerick Aubameyang's inviting delivery.