Sebastian Hoeneb's team initially struggled to find their rhythm against a high-pressing Augsburg side, who enjoyed the better of the early stages without creating any clear-cut openings.
With 26 minutes gone, Deniz Undav was denied from close range by Augsburg goalkeeper Nediljko Labrovic after running on to a magnificent Angelo Stiller pass into the penalty area.
Just four minutes later, however, Stuttgart were ahead after a defensive error from the visitors. Frank Onyeka's attempted clearance from a long ball struck Undav and landed at the feet of Demirovic. The VfB striker returned the ball to Undav, who sent a low shot into the bottom left corner.
The hosts grew into the game after that, striking the outside of the post in the 45th minute through Chris Führich before going into half-time with a deserved 1-0 lead.
Hoeneß's charges hit the ground running after the restart and dominated the early stages of the second period. Augsburg, however, also re-emerged with greater energy as the game developed into a hard-fought contest.
Undav sent an effort over the bar from the edge of the area in the 70th minute, before Alex Nubel saved a powerful long-range drive from Mert Komur just 60 seconds later. In an open and nail-biting end to the game, VfB's team spirit, work ethic and will to fight for one another helped them defend their narrow advantage and book their place in the last four of the DFB Cup.
Germany coach Julian Nagelsmann has urged Bayern Munich's 17-year-old talent Lennart Karl to focus on development with the under-21 side before thinking about a senior national team call-up.
Bayern Munich's teenage winger Wisdom Mike has played down the growing media attention surrounding his rapid rise, insisting he is focused purely on improving his game and earning his place at the club.
VfL Wolfsburg have sacked head coach Paul Simonis after a woeful spell of six defeats in their last seven Bundesliga matches, leaving the club in 14th place and just one point above the relegation playoff spot.
Bayern Munich were held to their first draw of the season by Union Berlin whilst Bayer Leverkusen hammered Heidenheim 6-0 in the Bundesliga on Saturday.
