They also said three people had carried out the attack on the Haripur Shia mosque Al Mostofa.
The shooting left Muezzin Moazzem Hossain, 70, dead and Imam Shahinur Rahman, 60, and Taher Mistri, 50, and Aftab Ali, 40, injured.
Moazzem’s uncle Ukil Mia told reporters, “The firing began when everyone was performing the Maghrib prayers.”
An eyewitness told bdnews24.com, “Before we realised what was happening, three young men opened random fire and fled by crossing the wall.”
Several people who had gathered for the evening prayers said one of the assailants wore a panjabi and a cap, while the other two had check shirts and trousers on.
He said about 30 devotees were performing the prayers when the firing started.
Ukil Mia said: “We have been left speechless by the indiscriminate firing. There has never been any Shia-Sunni conflict here. There was no quarrel ever over any mosque.”
Neither police nor local residents have been able to identify the attackers.
The Islamic State had claimed responsibility for the last month’s attack on the Shia procession in Dhaka but the government has repeatedly denied their presence in Bangladesh.
Bogra Superintendent of Police Md Assaduzzaman told newspersons: “This is an attack by criminals.”