Both teams have 21 points from nine games but Leverkusen are top on goal difference.
Schalke 04, who won 2-1 at struggling VfB Stuttgart, stayed on their heels on 19 points while Bayern Munich notched their first victory in three games with a 2-1 triumph at Freiburg that moved them up to fifth on 15 points.
The home team's injury woes continued when striker Tolgay-Ali Arslan limped off on 37 minutes, with forwards Paolo Guerrero and Mladen Petric already out with long-term problems.
Bruno Labbadia's Hamburg refused to give up though, surging forward again and going close through Eljero Elia and Marcus Berg.
Tunay Torun, 19, who came on for Arslan, missed the best chance for Hamburg when he failed to hit the target 11 minutes from time before Leverkusen's Sami Hyypia and Stefan Kiessling forced Rost into saves.
"We had to set the pace as Leverkusen waited for our mistakes," Labbadia told reporters. "We invested a lot but could not get the breakthrough because it's hard to replace players like Guerrero and Petric."
Schalke did the bare minimum to edge past Stuttgart with Ivan Rakitic taking advantage of Stefano Celozzi's failure to clear a cross from Rafinha to grab the opening goal in the 24th minute.
Schalke's Kevin Kuranyi then fired the winner three minutes later after Rafinha found him unmarked on the right.
