The police said officers fired at the 30-year-old, who was then taken to hospital.
According to information from DPA, she succumbed to her injuries there although the police initially did not confirm this.
The police said the woman initially injured a man with a knife in the area behind the Theresienwiese, home to the world famous Oktoberfest.
Shortly afterwards, she injured another woman with the knife.
The condition of the two was not immediately known.
It was also unclear whether the woman had any connection to them.
Several patrols were dispatched to arrest the woman, who continued to wield the knife.
Police then opened fire.
The number of shots fired could not initially be determined, according to police.
The woman was taken from the scene to a Munich hospital.
There was no danger to the general public, police said in a brief initial statement to the media in the evening.
The case brings back memories of an incident at Hamburg's main train station about two weeks ago: a woman randomly stabbed people on a platform for long-distance trains that was crowded with passengers.
According to police, 18 people were injured in the attack.
Officers arrested the 39-year-old woman and a magistrate ordered her placement in a psychiatric clinic.
