Two officers, a woman and one suspected militant were killed, while at least 12 other people were injured, officials said.
At least one of the bombs exploded during the prayer attended by hundreds of thousands of people at the sprawling Sholakia grounds in Kishoreganj, about 60 miles north of the capital, Dhaka.
It is the country’s largest open-air gatherings for the Muslim holiday of Eid-al-Fitr, marking the end of the fasting month of Ramadan.
After the blast killed two officers, police fired on the attackers and killed one of them, Assistant Superintendent Tofazzal Hossain said.
A female bystander was also killed in the crossfire. More than a dozen other people were injured with gunshot or bomb shrapnel wounds, including two in critical condition, he said.
Police cordoned off the area and searched devotees as well as nearby houses for suspects, said resident Shafiqul Islam, who was among those offering Eid prayers.
The violence comes just days after a deadly hostage crisis in which 28 people were killed, including 20 hostages, two police and six of the attackers.
Most of the hostages killed during the attack on a Dhaka restaurant on Friday were foreign - from Italy, Japan and India - raising international concerns about escalating extremist violence in Bangladesh.
The ongoing spate of attacks begun in 2013 has generally targeted atheists, religious minorities and others considered by militants to be “enemies of Islam”.
The country’s information minister said the target of the attack was the police convoy patrolling the religious gathering.