The match ended 1-1 at the Stade Velodrome in Marseille. The fans clashed after the match, injuring at least 35 people, reports Tass.
European football's governing body UEFA later started investigations against Russia for crowd disturbances, setting off of fireworks and making racist insults and is expected to impose a heavy sanction.
"I welcome the UEFA decision to impose sanctions on the teams whose pseudo fans have committed crimes. There should be no clemency towards them from sports federations and sports bodies," Cazeneuve said on Sunday.
"It is absolutely necessary that national football federations of those countries whose fans provoked disturbances are punished proportional to the seriousness of those incidents of which these people are accused of both inside the stadiums and outside them, what is more serious," the minister added.
Cazeneuve said the events in Marseille on Saturday are "absolutely unacceptable" both for the society, for the country's authorities and for those who really love football.
He also rejected criticism against police, saying the interior ministry is taking all the measures to ensure security at matches.
During the game, the Russian fans fired two flare guns and after the final whistle, the group rushed to the English fans' section.