The hosts dominated the first half, but failed to make countless chances count.
HSV then sparked into life with Cleber's goal, before Ivo Ilicevic and Nicolai Müller sealed the win.
Hungry Hannover unfortunate not to lead
Given their current situation, Hannover could have been forgiven for starting the game with an air of caution. However, Thomas Schaaf is known for his attacking approach and his team ploughed forward in the face of adversity. Hiroshi Kiyotake was, seemingly, trying to spark their revival on his own.
His chipped pass fell into the path of Adam Szalai whose powerful effort was well held by René Adler in the HSV goal. There was more concern moments later when a clever ball from Hiroki Sakai narrowly evaded the Hungarian's attempt to direct his cross goalwards.
While an interesting battle was developing down the flank of the two Sakai wing-backs, it was Hamburg who had the best chance of the opening stages. Sven Schipplock did excellently to maintain control of a loose ball that seemed destined to go out of play, before picking out Ivo Ilicevic. Somehow, however, he fired over an open goal.
Hannnover continued to in numbers, with Kiyotake picking passes and pulling the strings in midfield. Hugo Almeida was denied an opening goal when Adler stuck out a hand to flick his low, powerful shot around the post. HSV were backed into their own half and found it difficult to break the continuing pressure from their hosts.
The final chances of the half were distributed evenly. A Ceyhun Gülselam header looked to be heading for the bottom corner before Adler intervened, while it was a similar case at the other end. Hiroki Sakai made a vital block to prevent Schipplock from finding Nicolai Müller.
HSV pick Hannover apart
Gideon Jung was brought on at half-time for Lewis Holtby by Bruno Labbadia, in what seemed to be a move to try and curtail the influence of Kiyotake. That seemed to do the trick, as the Japanese was rarely seen in the first 10 minutes. Adler only had to gather an Iver Fossum volley, as HSV began to grow into the game.
Müller fed Ilicevic just before the hour mark, as the Croat went one-on-one with Ron-Robert Zieler. He opted to go high instead of passing round the goalkeeper, with Zieler standing tall to parry over the crossbar - his first real save of the game.
That relief would only be momentary, however. A corner straight from the training ground saw players pull away to leave Cleber with acres of space to dart forward and meet the delivery from Aaron Hunt. The Brazilian had stole a march on his marker and placed the ball powerfully past Zieler, which gave his side a vital lead.
What was seemingly a sick, reoccurring nightmare for the hosts showed no signs of ending when they fell further behind. Once again, a good start was not capitalised upon and they were left to rue missed chances. Ilicevic put the game beyond doubt with a tap-in from close range, following excellent build-up play from Schipplock.
Two minutes later, it seemed like the fate of Hannover was sealed. Schipplock was the creator for the second time in quick succession, feeding through Müller on goal. The winger rounded the helpless Zieler and sent the ball into the empty net; the home fans could hardly believe their luck.
Despite a late surge from the hosts and two more superb Adler saves, it was all in vain. HSV move to 10th, seven points clear of the drop zone and 17 ahead of Hannover. Schaaf's side remain rooted to the bottom of the table; their chances of beating the drop are slowly slipping away.