The man was hit in the leg by an officer on Sunday afternoon and is receiving medical treatment, a police spokesman said.
The suspect is a 39-year-old German, the DPA news agency has learned.
The incident raised major alarm as Hamburg is one of 10 German cities hosting Euro 2024 football matches.
Police say officers in Hamburg shot a man alleged to have been threatening people in the leg. (AP PHOTO)
Poland squared off there against the Netherlands on Sunday.
But a police spokesman said that early indications suggested "no football connection" and that the man acted alone.
The police operation was just a short walk away from the city's official fan zone.
Police said football fans there were "well-protected" and never in danger.
Tens of thousands of Dutch supporters earlier marched down the streets of the St Pauli nightlife district, where the Reeperbahn is located.
According to police, the man had come out of a pub on a street adjacent to the Reeperbahn with what was described as a pickaxe.
He waved the tool at police officers in a "threatening manner," according to the spokesman's account.
He was also carrying a Molotov cocktail.
Police asked him to put down the axe.
"He put down the pickaxe and then tried to set fire to the Molotov cocktail," a police spokeswoman said.
After several warnings, the officers opened fire and shot him in the leg.
Video footage circulating on social media showed the man threatening several people.
He tried to climb a barrier in front of several police officers.
He can be heard yelling at the police officers to "shoot".
The man can be seen running towards other people and tried to ignite the Molotov cocktail.
Officers initially used pepper spray before several shots are fired.
The crime scene was cordoned off by police on Sunday afternoon.
Police said no other people or officers were injured.
