Hannover needed a penalty shoot-out to see off the second division side on a nervous night in Lower Saxony.
"We knew that we were going to need something special if we were to go through," Hannover coach Mirko Slomka told Sky television.
"We have now got four games in our legs and you could see that, particularly in extra-time.
"If we had gone out, it would have got very uncomfortable for us in the next few days, but this way, we can go into the Augsburg game with a lot of self-confidence.
"We now have a bit of time to recharge our batteries."
Augsburg had a slightly more comfortable passage into the last 16 of the cup with a 1-0 victory in Westphalia.
"We scored the decisive goal and have reached our goal," said their coach Markus Weinzierl.
"We are pleased because we are into the next round, even if it was not spectacular.
"The win has given us more self-confidence. Hannover are very strong in attack and we are going to have to be at our best in defence and take our chances better up front."
Weinzierl could have influential midfielder Koo Ja-cheol back in his squad after the South Korea international returned to training this week while Torsten Oehrl has recovered from the flu, although Dominik Reinhardt is ruled out after an operation on his toe.