Hamburg made it 10 points in their last four games with victory Greuther Furth, who slumped to their fourth home defeat.
Son Heung-min decided the game in the 17th minute with the Bavarians unable to find a response in a tense encounter
Hamburg extended their unbeaten record to four games in which they have lifted themselves from the foot of the table to eighth, level on points with fifth-placed Hannover.
The game started slowly before South Korean midfielder Son put the visitors in front in style.
He was able to dribble unchallenged for almost 30 yards before sending a low shot past Max Grun and into the bottom corner of the net.
Furth looked devoid of confidence and Mike Buskens' side struggled to find an answer in a game which was frequently interrupted by fouls and yellow cards.
Gerald Asamoah had the home side's best chances, but Hamburg goalkeeper Rene Adler, who passed a late fitness test, had very little to do.
Indeed, the former Germany international's condition was never truly tested by the timid hosts, not even when an extra striker in Christopher Nothe was introduced.
Instead, the best chances arrived at the other end with Artjoms Rudnevs twice failing to find the net.
His second big chance, when his effort hit the outside of a post, drew an angry reaction from Rafael van der Vaart, who was waiting to for a tap-in in front of an open goal, the Latvian forward instead trying to score from a tight angle.
It did not matter, though, as Furth never came close to finding an equaliser, allowing Hamburg to carry on their excellent recent form.