Hamburg, the only top flight club never to have been relegated since the league's creation in 1963, suffered a setback when keeper Rene Adler injured his back during the warmup and Jaroslav Drobny came in to replace him.
With a sold-out crowd of 57,000 and their iconic stadium clock marking the 50 years 264 days and five hours that the team has spent uninterrupted in the Bundesliga, Hamburg struggled against the best attack in the second division.
"They did it really well and had us under control especially in the first half," Hamburg coach Mirko Slomka told reporters.
"They were extremely dangerous up front. But we still have one game left and Fuerth will be nervous at home."
It was Fuerth forward Ilir Azemi who had the first chance, but the 22-year-old's shot from a tight angle was saved by Drobny.
With experience in promotion battles, Fuerth, who were relegated last season, were more aggressive. Hamburg had to wait until the 44th minute for a chance with Hakan Calhanoglu's shot sailing wide.
Fuerth picked up where they left off after the break with Nikola Djurdjic missing a golden chance when he narrowly failed to connect with a perfectly-timed cross in the box.
Hamburg upped the tempo after an hour with forward Pierre Michel Lasogga going close on two occasions as the hosts were urged forward by their fans, who grew more desperate as the minutes ticked away.
"We played a very good game today and we had good chances but we did not bite," said Fuerth coach Frank Kramer. "We have to do it on Sunday then."
The two sides will meet again on Sunday in the return leg, where away goals count double in the event of a draw.
Nuremberg and Eintracht Braunschweig were relegated from the Bundesliga, with Cologne and Paderborn securing automatic promotion to the top flight.
- Sports & Recreation
- Jaroslav Drobny