Furth took the lead in the 14th minute through Nikola Djurdjic but Maximilian Beister equalised for Hamburg in the 22nd minute.
Last-placed Furth had failed to score in five of their last six games but at least managed to do that, however with only 14 points their position may be irretrievable. Hamburg moved to 35 points and watched Schalke pass them in the table to reach sixth place.
Hamburg failed to convert two early chances as Marcell Jansen headed just over the crossbar in the first minute and Furth goalkeeper Wolfgang Hesl made a fine save from Beister two minutes later.
The visitors used their first opportunity to take the lead as Thomas Pledl sent a looping pass over the Hamburg defence and Djurdjic's shot beat Hamburg goalkeeper Rene Adler.
Hamburg equalised when Dennis Aogo passed to Beister who turned and blasted into the upper left corner with his left foot from 14 yards.
Hesl kept his team in the game with a superb stop to thwart Beister's attempt from 17 yards five minutes later.
Hamburg had another chance after 31 minutes but Rafael van der Vaart's shot went just inches wide left.
Van der Vaart was sent towards Hesl in the 50th minute by Slobodan Rajkovic but the Dutchman could not beat the goalkeeper.
Furth had a chance six minutes later but Stephan Furstner shot just wide to the right of goal from the edge of the box. And Furstner was again just inches off target with an attempt from 18 yards in the 69th minute.
Hamburg missed a golden chance to grab the win with six minutes remaining when Marcus Berg shot wide to the left of the target despite having plenty of space.
And Hesl saved Furth an important point by denying Heiko Westermann a minute into injury time.