Mirko Slomka's men were in danger of being demoted from the Bundesliga for the very first time on Sunday following a tumultuous campaign.
The club finished the league season with five successive defeats and ended up finishing just one point ahead of Nuremberg, who suffered automatic relegation.
Furth, on the other hand, went into the two-legged play-off in fine form and gained extra confidence from a goalless draw in the first leg at the Imtech Arena.
However, it was Hamburg who claimed an early advantage in the return match, as Pierre-Michel Lasogga headed home a Rafael van der Vaart corner in the 14th minute.
The visitors suffered a blow on the half hour when Johan Djourou was carried off after damaging a vertebra in his neck, an injury that puts his participation for Switzerland in the FIFA World Cup in doubt.
Stephan Furstner then equalised for Furth after 59 minutes, setting up a tense finish that saw Hamburg come under increasing pressure.
There was to be no third goal, though, and following a brief fracas at full-time, Hamburg were able to enthusiastically celebrate a valuable away-goals triumph.