Similar violence was also reported in Bangladesh's second-largest city, Chittagong, which is in the southeast.
In Dhaka, opposition supporters allegedly set fire to a car and a bus, but no injuries were reported.
The government has banned rallies in some parts of the country, fearing violence as the opposition gears up for street protests. Authorities deployed paramilitary border guards in Dhaka and other cities and towns to aid police in maintaining order.
The violence came as opposition leader Khaleda Zia said Friday that she and her allies would stage a three-day general strike across the country beginning Sunday morning to back demands for a caretaker government made up of people from outside political parties to oversee elections that are due by early January.
Zia told a huge rally that Prime Minister Sheikh Hasina had two days to agree to the demand, otherwise more protests will be announced. Street protests in Bangladesh often turn violent.
Hasina recently proposed establishing an all-party government with herself as the leader to oversee the elections, but Zia, a former prime minister, rejected that.
Hasina offered to hold talks in line with her proposal, but Zia refused.
Such a caretaker government has been used in the past, but Hasina's government scrapped the provision in 2011, citing a Supreme Court ruling saying the provision was unconstitutional. But the opposition says the dropping of the system makes it easier to rig the election a charge denied by the government.
A senior ruling party leader, Mohammad Nasim, criticized Zia, saying she has picked a "path of confrontation, not of dialogue." He also urged Zia to agree to Hasina's proposal.
Bangladesh has had a parliamentary democracy since Hasina and Zia together threw former military dictator H.M. Ershad out of power in 1990, but peaceful transfers of power have remained a big issue.