The break came at a good time for the visitors, who dominated in the early stages of the second half as Miiko Albornoz forced Christian Mathenia into a good save at the near post, before Sane headed over from a corner.
It soon became clear that neither side were going to settle for extra time, and it was Darmstadt who finally made the breakthrough with 16 minutes of normal time remaining.
Poor marking from a free kick cost Hannover as a free kick somehow reached Aytac Sulu, who converted from close range to hand his side control of the tie.
However, an equaliser arrived less than a minute later as Artur Sobiech capitalised on hesitancy in the box to pick out the bottom corner with a fine finish.
The drama was not over, though, with Sandro Wagner securing his side's progress in the competition by firing home from a Tobias Kempe corner as more poor defending from the visitors made the difference.