Gilberto Silva and substitute Tomas Rosicky scored for the last season's runners-up before Boubacar Sanogo notched the consolation goal for Hamburg.
The German side played with ten men since the 11th minute after goalkeeper Sascha Kirschstein was sent-off for bringing Robin Van Persie down inside the area.
The victory put the Gunners in charge of the group with three points and left Hamburg rooted.
Porto and CSKA Moscow are second and third with one point each after their clash ended in a scoreless draw.
Emmanuel Adebayor launched an early raid from the left flank and set-up Van Persie who was clipped in the box by Kirschstein.
Arsenal were rewarded a penalty-kick and Hamburg's starting goalkeeper was handed a red card.
Brazil's World Cup winner Silva, who captained Arsenal in the absence of injured Thierry Henry, converted the penalty home as Arsenal grabbed the lead and enjoyed a numerical advantage.
Rosicky doubled the lead eight minutes into the second period with a trademark shot from outside the area that settled into the back of the net.
Sanogo pulled one back in the stoppage time when he converted a Piotr Trochowski low pass into Jens Lehamnn's net.