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The confrontation took place in the Seans Peña plaza, near the Maracana stadium about an hour before Argentina played Germany. Fans inside the stadium seemed unaware of the protest.
The military police spokeswoman declined to give further information but local media said about 200 protesters in the area had tried to make their way towards Maracana.
Protests across Brazil have petered out since the month-long soccer tournament started on June 12, but authorities are trying to preempt violence during the final now that Brazil is out of the tournament and online videos have encouraged violent fringe groups to return to the street.
Police in Rio on Saturday detained 19 people with a history of committing vandalism during protests, earning sharp condemnation from human rights groups that said the detentions were unwarranted.
A police crackdown against students a year ago was considered largely responsible for setting off a protest movement that brought more than a million people into the streets to air grievances over corruption and rising prices and to contrast the high cost of the tournament with the poor state of public services.
About 26,000 security forces were on duty during the final match attended by several presidents, while 1,500 private security guards were inside Maracana.