No one immediately claimed responsibility for the attack.
The explosions occurred late on Friday at Akhaura in Brahmmanbaria district, 80km east of the capital, Dhaka, as people were taking part in an annual festival, area police chief ATM Tareq said.
An unidentified man died instantly during the attack at the Hazrat Shah Syed Ahammed shrine, Tareq said.
About 40 of the injured were being treated in hospitals while another 10 were released after receiving first aid.
Police and residents said thousands of visitors, including women and children, attend the festival and a week-long fair each year.
The reason behind Friday's attack was not immediately clear.
"We need a detailed investigation to find out the actual reason," Tareq said.
In 2004, bomb attacks at Bangladesh's shrines separately killed about a dozen people.