He then careered towards the pitch, narrowly missing some spectators in the stands before destroying technical equipment and eventually landing on the grass.
Joachim Herrmann, Bavaria’s interior minister for the state, said marksmen spotted the protester's Greenpeace banner and therefore decided not to intervene.
Footage of the incident showed that the activist's parachute had the words "Kick out oil Greenpeace" written across it.
“But if the police had come to a different conclusion that they might have been dealing with a terrorist attack, the aviator may well have paid with his life,” Mr Herrmann added.
Munich police said two people were taken to hospital for what they described as non-serious injuries.
Officers have arrested a 38-year-old man from Baden-Württemberg state in connection with "various offences".
The police force said in a statement: "In this connection, investigations are now being conducted for various offences under the Criminal Code and the Air Traffic Act, among others.
"The Munich Police Department emphasises that it has no understanding whatsoever for such irresponsible actions, in which a considerable endangerment of human life is accepted."
Environmental group Greenpeace said the activist had only meant to drop a balloon inside the stadium to raise awareness of the climate crisis before he encountered technical difficulties.
"We ask the two people injured yesterday during the Greenpeace action to accept our sincere apologies," a statement from the group said.
"The pilot wanted to float a balloon into the stadium but then had to make an emergency landing because the manual throttle control of the paraglider failed."
Speaking after his side recorded a 1-0 victory over Germany, France manager Didier Deschamps condemned the incident, saying it should never have happened and that a tragedy had been avoided.
"So, what can I say? It shouldn't have happened," he said. "But at the same time, when you witness events like this... it will become literature because there were no consequences."