The polls have been boycotted by the opposition and described as flawed by the international community.
Prime minister Sheikh Hasina's refusal to heed opposition demands to step down and appoint a neutral caretaker to oversee the election led to the boycott, undermining the legitimacy of the vote.
Opposition activists have staged attacks, strikes and transportation blockades amid unrest that has left at least 286 people dead since last year.
"We never expected such an election," said Aminul Islam, a resident of the capital, Dhaka, who refused to vote.
"For such a situation, both the government and opposition are responsible. They don't want to establish democracy."
Police opened fire to stop protesters from seizing a polling centre in Bangladesh's northern Rangpur district, killing two people, authorities said.
In a similar incident in neighbouring Nilphamari district, police fired into about two dozen protesters, killing two people.
Police gave no further details, but Dhaka's Daily Star newspaper said the four men who were killed belonged to the opposition Jamaat-e-Islami party.
Police said another seven people were killed in violence elsewhere, including a polling official who was stabbed to death by suspected opposition activists. At least 127 polling stations were attacked in overnight attacks.