The assailant entered the supermarket in Barmbek district on Friday afternoon and stabbed customers before fleeing, police said.
Witnesses followed him and alerted police, who arrested him near the scene.
Police said several people suffered stab wounds and one of the victims was fatally wounded.
There was no information on the assailant’s motive or on the exact number of victims.
Police said there was definitely only one attacker. It added that “initial reports about robbery as a possible motive so far have not been confirmed.”
On December 16, 2016, a Tunisian hijacked a truck, driving it into a crowd shopping at a Christmas market in the German capital, Berlin, in an act of terror that left 12 people dead.