A second match in the northern town of Gradacac was interrupted for 40 minutes because of nationalist chanting, they said.
In the capital, Zeljeznicar Sarajevo fans fought with each other and then clashed with police in the streets after the club's 2-0 win over visiting Travnik.
"Zeljeznicar supporters first fought amongst themselves and then attacked the police," Sarajevo police spokesman Irfan Nefic said.
"Three people were detained over the weekend and police will press criminal charges against them," he added. "Prison terms for attacks on officers on duty range from six months to five years."
The Gradacac game on Saturday was disrupted after visiting Borac Banjaluka fans from the Bosnian Serb Republic, one of the ethnically-divided country's two autonomous regions, shouted nationalist slogans and clashed with home supporters.
The game ended in a 1-1 draw after police escorted Borac fans, who sang Serb nationalist songs to provoke home supporters from the city in Bosnia's Muslim-Croat Federation, out of the stadium.
The Bosnian Football Federation said it was considering sanctions against Borac's supporters.
"We may introduce a stadium ban just for Borac fans, who often cause incidents," the federation disciplinary committee's president Meho Alibasic said.
Last April, a cup tie between Borac and Zeljeznicar was abandoned after Borac fans hit a linesman with a missile, and in September 2011, a league match between the clubs was cut short when Borac fans invaded the pitch and hurled rocks and flares at visiting Zeljeznicar supporters.
- Sports & Recreation