Hertha continued to look the more dangerous of the two sides as the interval approached, with Kalou heading a difficult chance wide of the target shortly before Drobny pushed away a long-range strike from Anis Ben-Hatira.
The visitors enjoyed a spell on top early in the second half, but their inability to turn pressure into an opening goal cost them as Hertha struck the crucial first blow.
Valentin Stocker created the goal with a fine run before picking out Anis Ben-Hatira, who used the space on the edge of the box to pick out the bottom corner.
The lead was doubled six minutes later when John Heitinga reacted quickly to fire home from close range after Hajime Hosogai had headed against the post.
Hamburger struggled to produce any kind of response to falling behind, and a comfortable Hertha victory was completed late on when Ben-Hatira linked up with Per Skjelbred before adding his second of the afternoon with an emphatic finish.