The three-time Bundesliga winners, the only club never to have been relegated since the creation of the Bundesliga in 1963, got the golden goal through forward Pierre-Michel Lasogga in the 14th minute after the goalless first leg on Thursday.
"We did it with the narrowest of margins and in the most nerve-wracking way," Hamburg coach Mirko Slomka told reporters.
"To take the lead and then concede the equaliser in the second half, then you have the fans against you. It was tough and we have now used up all our luck. We were extremely lucky and should not count on having any next season."
Lasogga, who returned from injury just in time for the last game of the regular season and the playoff games after a six-week absence, made the most of the resulting corner, rising high to drill in a header.
The 22-year-old, who missed out on a Germany call-up due to his injury, almost doubled the lead early in the second half with a point-blank header but it was battling Fuerth who scored on the hour when Stephan Fuerstner slid in to equalise.
"I was out for six weeks so the last 20 minutes hurt a lot," Lasogga said. "But it does not matter how we did. We just did. We did not win the title today but stayed in the Bundesliga and that's what matters."
The scene was set for a nervous final half hour and it proved to be just that with Hamburg hanging on for the draw they needed and Fuerth desperate for a winner.
Fuerth striker Ilir Azemi should have done better when he found himself in the box in the closing minutes with a clear sight of goal but he miskicked and squandered his chance.
Hamburg managed to hold on, running out the clock to end a disastrous season on a high as Fuerth, arguably better over both legs, failed to return to the Bundesliga after being relegated last season.
Eintracht Braunschweig and Nuremberg were relegated with Cologne and Paderborn securing automatic promotion from the second division.
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- Pierre-Michel Lasogga