No group has admitted carrying out the attack at 2130 local time (1530 GMT) on Saturday, but Mr Nasim has blamed Islamic extremists.
The Home Minister, Mohammad Nasim, called the bombing a "heinous attack" as he visited the scene.
The injured include the local member of parliament, Shamim Osman.
He had been addressing party supporters at the offices in the port town of Narayanganj, near the capital Dhaka, when the blast occurred.
No group has admitted carrying out the attack at 2130 local time (1530 GMT) on Saturday, but Mr Nasim has blamed Islamic extremists.
The explosion happened during a meeting of the party ahead of a general election scheduled for October.
According to one report 14 people were killed instantly in the explosion, while the rest died in hospital or while trying to get there.
Political rivalries in Bangladesh are intense and often explode in violence ahead of elections.
Police say that the device was hidden under a table and that many people died immediately.
The tin roof of the office was blown off by the force of the explosion, a spokesman for the Narayangangj police said.
Television pictures broadcast in Bangladesh showed bodies badly burned or ripped apart by the explosion.
The emergency services are now making urgent appeals for blood donations.
Series of bombs
This explosion is the latest in a series of random bomb blasts in Bangladesh for which no group has claimed responsibility.
In June there was a bomb attack on a Roman Catholic church in which 10 people were killed.
Two months before that another bomb went off during a concert in the capital Dhaka to celebrate the Bengali new year - eight people were killed.
On that occasion the government had also blamed Islamic extremists.
The Awami League government of Prime Minister Sheikh Hasina is facing a strong challenge from the opposition Bangladesh Nationalist Party at the elections.
The prime minister is shortly expected to hand over power to a caretaker administration before the vote is held.
The government last week approved a law which proposes to give Sheikh Hasina life-long security.
It says that her life is under threat after the conviction of 12 former army officials found guilty of murdering her father, Sheikh Mujibur Rahman, who was the first president of Bangladesh.