Babbel was given a vote of confidence by Stuttgart's bosses only last week, but with his side now suffering their fifth consecutive defeat and hovering above the relegation zone in the league, Babbel's position is precarious.
"We wanted to win here and reach the next round, which we haven't done and the situation is not easy now," said Stuttgart's grim-faced director of sport Horst Heldt when asked about Babbel's position.
And Babbel is realistic about his situation.
"I do not know where I stand with the club now," he admitted. "If I hate one thing, it is to lose, but now we have five consecutive defeats - this is a bitter and difficult situation. It is not getting easier, but I still believe in this team," he said.
Stuttgart went behind on 32 minutes through midfielder Bernd Nehrig after teammate Leonhard Haas found him with a pin-point accurate pass and the 23-year-old's shot gave former Germany goalkeeper Jens Lehmann no chance.
With his job on the line, Babbel brought Russian play-maker Aliaksandr Hleb off the bench after 59 minutes, but the ex-Barcelona star only earned himself a yellow card on 70 minutes.
With time running out, both Romanian striker Ciprian Marica and 20-year-old replacement forward Julian Schieber went close, but could not prevent Stuttgart from slipping out of the German Cup, while Fürth are in the quarter-finals.
At the final whistle, Stuttgart's senior players like Germany midfielder Thomas Hitzlsperger were left shattered on the turf while Fürth's players celebrated.
Dortmund also crashed out of the cup when they were beaten 3-2 at third division Osnabrück, who had claimed Hamburg's scalp in the second round, after Italian midfielder Angelo Barletta scored twice for the home side.
Argentina striker Lucas Barrios scored his fourth goal in as many games deep in injury time, but his 96th-minute strike came too late.
But fellow Bundesliga outfit Cologne had no such problems as they saw off regional league side Eintracht Trier with a comfortable 3-0 away win to take their place in the last eight.
And Augsburg also booked their passage into the next round with a 5-0 hammering of fellow second division side Duisburg.
In Wednesday's other third round fixtures, Bayern Munich are at Eintracht Frankfurt, Schalke are at Munich 1860, Hoffenheim host Koblenz and Kaiserslautern are at Werder Bremen.