The end of the game was delayed as Hamburg fans set off flares causing a ten-minute delay.
Wolfsburg's 4-1 mauling of Cologne meant that Hamburg's 2-1 home win over Borussia Moenchengladbach on Saturday, wouldn't have been enough to kee them in the top flight.
Protests from Hamburg fans delayed the end of their season finale in some chaotic scenes, but they could not halt the Bundesliga club from suffering relegation for the first time in their history.
"This is an extremely painful moment for the entire Hamburg SV", club President Bernd Hoffmann said. "The club has lost one of the things that makes it unique".
Hamburg were leading 2-1 in stoppage-time when referee Felix Brych stopped the game.
There was to be no great escape for Hamburg on Saturday afternoon, with relegation to the second tier confirmed on the final day of the Bundesliga season..
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Glenn Maxwell, Shahbaz Nadeem and Liam Plunkett made in way for Abhishek Sharma , Sandeep Lamichhane and Junior Dala. With the target just a stone's throw away, Kohli (70) was caught by keeper Rishabh Pant off leg spinner Amit Mishra.
Hamburg had taken an early lead through Aaron Hunt before Gladbach's Josip Drmic levelled the score just before halftime.
Even when Hamburg went down to 10 men after the dismissal of Bobby Wood in the 71st minute, they kept battling, hoping for a miracle but events at Wolfsburg ensured it never came.
As the only club to have played in every Bundesliga season since its inception in 1963, supporters of the six-time German champions have proudly displayed a clock at Volksparkstadion which keeps a record of their top-flight status.
Bayer 04 Leverkusen might have pipped Dortmund to Europe's top competition on goal difference, but fell short in a 3-2 win over Hannover 96 and must be content with fifth place and Europa League qualification.
Leverkusen instead are forced to settle for a place in the UEFA Europa League, along with Leipzig, whose 6-2 thumping of Hertha Berlin ultimately counted for little as the other results failed to go their way.