The victims were travelling to the court office on Monday when unidentified assailants threw the bomb at them in Jhalakathi town, 120km south of the capital, Dhaka, local police official Sohrab Hossain said.
Sohel Ahmed, an assistant judge, died instantly, while another judge, Jagannath Pandey, died on the way to hospital, Hossain said.
Police arrested a man who had been injured at the scene, but no details were immediately available about him, and no organisation has claimed responsibility for the attack. The motive is unknown.
Hossain said a vendor was also injured, and both had been admitted to hospital.
The driver was unhurt because he left the vehicle minutes before to call another court official to take him to the office.
Hossain added that a portion of the roof of the minibus was blown away by the bomb and additional police have been deployed at government offices in the town.