Third division VfL Osnabrueck caused another upset when they beat Bundesliga side Borussia Dortmund 3-2 to book a spot in the last eight.
Stuttgart's defeat will pile further pressure on embattled rookie coach Markus Babbel whose team lie 14th in the league with only eight points from 10 games.
Bernd Nehrig put Fuerth ahead in the 32nd minute with the hosts' first chance, when he left Arthur Boka and Mathieu Delpierre standing to fire home.
They could have added another on 49 minutes when Sami Allagui found himself in the box but failed to make it 2-0 in this one-leg Cup match.
Stuttgart, unable to string together any real attacking moves, wasted a rare double chance 12 minutes from time when Julian Schieber and Romanian Ciprian Marica were denied from point-blank range by Fuerth keeper Stephan Loboue.
Marica also hit the post three minutes from time and Schieber missed a great chance to equalise from close range in the dying seconds.
Osnabrueck, who eliminated Hamburg SV on penalties in the last round, took the lead in the 37th with an Angelo Barletta bicycle kick.
The Italian central defender added another three minutes from halftime before Nuri Sahin pulled one back for Dortmund 10 minutes after the restart. Benjamin Siegert secured Osnabrueck's victory when he netted on a quick break in the 69th minute.
Cologne had little trouble seeing off fourth division Eintracht Trier with a 3-0 win.
Bayern Munich are in action against Eintracht Frankfurt on Wednesday when holders Werder Bremen take on Kaiserslautern.
