Stuttgart keeper Alexander Nuebel saved three spot kicks with a total of 20 penalties taken after the entertaining encounter had ended 4-4 after 120 minutes.
The holders fell behind in the eighth minute through Sven Koehler's shot but the visitors levelled three minutes later with Ermedin Demirovic heading in the equalizer.
The Bosnian then put them in front on the hour mark before a double strike by Fabio Di Michele Sanchez late in the second half turned the game around.
Stuttgart were rescued when striker Nick Woltemade snatched an 89th-minute equalizer to force extra time.
They thought they had made sure of their spot in the next round when Tiago Tomas spectacularly dribbled past opponents to cut the ball back into the box and force Sanoussy Ba to turn it into his own goal.
Braunschweig, however, came back once more, hitting the crossbar in the 99th minute and bagging another equaliser through Christian Joe Conteh's quick turn and shot in the box in the 105th.
Nuebel then saved three of the 10 penalties taken by Braunschweig, with Stuttgart's Lorenz Assignon scoring the decisive spot kick.