An Italian police officer shot Amri dead near Milan's Sesto San Giovanni train station on Friday.
"The Berlin attacker carried out a new attack against an Italian police patrol in Milan and was killed in an exchange of fire," the IS-linked Amaq agency said, referring to the fatal shooting of Anis Amri, the prime suspect in the Berlin attack, by Italian police.
Twelve people were killed when the Polish-registered articulated truck, laden with steel beams, slammed into the crowded holiday market late Monday, smashing wooden stalls and crushing victims.
Amaq said Tuesday that "a soldier of the Islamic State" had carried out the attack "in response to appeals to target citizens of coalition countries".
Germany is part of a US-led coalition fighting IS in Iraq and Syria.
An Italian police officer shot Amri dead near Milan's Sesto San Giovanni train station on Friday, four days after the attack which had triggered a Europe-wide manhunt.