The incident happened in the southern city of Wangen when two officers attempted to detain the 27-year-old man at his home to take him to serve a prison sentence for assault.
"In the course of the operation, the man suddenly pulled out a knife and attacked the officers without warning," police and prosecutors said in a joint statement.
"As a result of the attack, the police officers fired several shots."
The Afghan died at the scene despite efforts to resuscitate him, they said.
One officer was seriously injured after being stabbed several times but was not in a life-threatening condition.
The authorities have launched an investigation into the use of a firearm by police.
Migrants have carried out several serious attacks in Germany in recent times, including knife attacks and car-rammings, fuelling a heated debate on immigration.
Chanceller Friedrich Merz, who took power in May, has launched a crackdown on irregular immigration, which includes turning away most undocumented migrants at Germany's borders.
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