At Hamburg - who had survived the relegation play-offs in 2014 and 2015 - Gladbach fans mocked their counterparts by unfurling their own clock which ticked down towards their opponents' impending demotion, and as the game entered stoppage time a number of home fans hurled flares towards the pitch.
Jonas Hector brought Cologne level just after the half-hour mark with an exquisite chip, but Divock Origi restored the lead for Wolfsburg, edging Hamburg closer to disaster.
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.
After a 15-minute delay due to crowd trouble, the game was able to end.
HSV had played in the German top flight since their foundation and in the Bundesliga since it began in 1963 but, after several flirtations with the drop in recent years, they have finally bowed out.More news: Himanshu Roy has left a void hard to fill in: Mumbai Police
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Elsewhere, Hoffenheim and Borussia Dortmund joined Schalke and champions Bayern Munich in the league's top four.
Dortmund goalkeeper Roman Buerki gifted Hoffenheim possession with a awful clearance on 26 minutes, allowing Andrej Kramaric to fire the home side in front.
Andrej Kramaric and Adam Szalai scored for the hosts, either side of Marco Reus' equaliser, and when Pavel Kaderabek netted Hoffenheim's third, Leverkusen were 3-0 up at home to Hannover needing a further two-goal swing to usurp Dortmund and snatch fourth.
Lucas Alario grabbed a brace as Leverkusen eased to a 3-2 win over Hanover, as both they and Leipzig secured a Europa League place.
Bayern, though, were unable to give coach Jupp Heynckes an appropriate send-off in his final home game, as they stumbled to a shock 4-1 defeat to Stuttgart.