The suspected attack also came hours before international leaders including US Vice President JD Vance and Ukrainian President Volodymyr Zelenskiy were due to arrive in the city for the Munich Security Conference.
Police said a white car had approached police vehicles that were accompanying a demonstration of striking trade union workers, before speeding up and hitting people.
Officers arrested the suspect after firing a shot at the car, deputy police chief Christian Huber said.
Officers detained the 24-year-old driver.
His motive was unclear.
"It was probably an attack," Bavaria state premier Markus Söder told reporters.
Bavarian interior minister Joachim Herrmann said the man had not previously been found guilty of shoplifting, contradicting statements made earlier in the day.
The suspect was a 24-year-old Afghan asylum seeker, Huber said.
He came to Germany at the end of 2016 as an unaccompanied minor refugee.
His asylum procedure was finalised in 2020 with a rejection notice and an order to leave the country.
Söder called the apparent attack "a punch in the face" for Germany and said that there must be consequences once authorities determine exactly what happened.
The incident follows a series of attacks involving immigrants in recent months that have pushed migration to the forefront of the campaign for Germany's February 23 election.
In December, six people were killed in an attack on a Christmas market in Magdeburg and last month a toddler and adult were killed in a knife attack in the Bavarian town of Aschaffenburg.
Immigrants have been arrested over both attacks.
Conservative Friedrich Merz, frontrunner to be Germany's next chancellor, said safety would be his top priority.
"We will enforce law and order. Everyone must feel safe in our country again. Something has to change in Germany," Merz posted on X.
Social Democrat Chancellor Olaf Scholz said the perpetrator could not hope for leniency.
"He must be punished and he must leave the country," said Scholz, according to news outlet Focus Online.
"If it was an attack, we must take consistent action against possible perpetrators with all means of justice."
About four hours after the incident, the street where it took place was strewn with items of clothing and bags, a shoe and a pair of glasses.
Police set up a gathering point for witnesses in the Loewenbraeukeller, one of Munich's oldest beer halls.
A passer-by said he witnessed the incident from a window of a neighbouring office building.
The car, a white Mini Cooper, had threaded its way between the police vehicles and then accelerated, he said.
Another witness said she had seen part of the incident from a building.
The car had accelerated and hit several people in the crowd, she said.
People in the crowd had been taking part in a strike held by the Verdi public sector workers' union whose leader, Frank Werneke, expressed shock but said he had no further details.
Bavaria's interior minister said he did not suspect there was a connection to the Munich Security Conference, which starts on Friday.
with Reuters and DPA
