The man injured the face of an officer at the entrance to the police station in Joué-lès-Tours near the city of Tours in central France and injured two other before he was shot, the interior ministry said.
Anti-terrorist investigators were probing the incident, a judicial source said.
The perpetrator was a French citizen born in Burundi in 1994 and known to police for common crimes, a source close to the case told AFP, adding that the attacker "shouted 'Allahu Akbar' from the moment he entered until his last breath."
According to a statement by the interior ministry, the assailant was around 20 years of age, and was "killed (by) police officers present using their issued firearms."
The attacker was not on any watch lists maintained by France's main domestic intelligence service, the General Directorate for Internal Security, the source involved in the inquiry said.
But the source noted the assailant's brother was known to security agencies for his radical convictions and had at one point planned to travel to Syria.
"The investigation is leading towards and attack on police forces on radical Islamist motives," another source close the case said.
Officials said the condition of the three wounded police officers was stable.