Police used water cannon and rubber bullets to break up the fighting between about 100 reputed Polish hooligans and a much larger crowd of Russian fans marching to the National Stadium in Warsaw.
"A group of hooligans tried to stop the march by Russian supporters," police spokesman Maciej Karczynski told the Polish Press Agency.
Ten people were hurt in the scuffle, which preceded a 1-1 tie by the two teams in their Group A match.
The president of the Russian Soccer Union, Sergei Firchenko, told the ITAR-TASS news agency only a few Russian fans were among those arrested.
"As far as I know, they behaved well during the march," he said. "Polish fans have much bigger problems."
Russia's RIA Novosti news agency said about 5,000 Russian fans were marching to commemorate the Russia Day holiday -- which commemorates the end of the Soviet Union -- as much as to get fired up before the game.
The Russian fans had earlier been chastised in Polish media because some of them waved the old USSR hammer-and-sickle flag during a match against the Czech Republic, RIA Novosti said.