The Tunisian criminal fired on Italian police who asked him for ID during a routine patrol in the Sesto San Giovanni area in the early hours of Friday.
German authorities say fingerprints they provided have confirmed the dead man is Amri.
Monday’s attack on a Berlin Christmas market left 12 people dead and 49 injured.
German Chancellor Angela Merkel said she was “relieved” that the attacker had been neutralised but added that Islamist terrorism was “a recurring threat to us”.
Meanwhile, the Islamic State group has released a video showing Amri pledging allegiance to its leader, Abu-Bakr al-Baghdadi.
He does not make any reference to the Berlin attack, which Isis claimed soon afterwards. It is not clear when or where the video was filmed.
Shortly before releasing the video, Isis acknowledged Amri’s death in Milan.
When Italian police stopped the suspect, who was on foot, he “immediately drew out a gun” and shot at the two policemen, Italian Interior Minister Marco Minniti said.
Officer Cristian Movio was injured in the shoulder but his injuries are not life-threatening.
His junior colleague, Luca Scata, who had been in the police for just nine months, was the one who fired the shot which killed Amri.
- BBC