Mizanur Rahman Bablu, a local leader of the Bangladesh Nationalist Party, died from wounds to the chest on his way to a hospital after the blast late on Sunday in Khulna city, 136km southwest of the capital Dhaka, police official Razaul Karim said.
Bablu was sitting with three friends in a shop after nightly prayers when two unidentified men hurled the homemade bomb at them, Karim said.
No one claimed responsibility for the attack and police said they were investigating the motive.
Bangladesh has been hit by a spate of bombings that have killed four and injured dozens of people across the country in August and October.
Police arrested and questioned more than 500 suspects.