District vice president Sheikh Mohammad Habibullah was killed when a bomb was hurled at a rally in the eastern Brahmanbaria district, area police chief Masudul Alam said.
A man also died of bullet wounds on Friday, a day after he was caught in clashes between members of the ruling Bangladesh Nationalist Party (BNP) and supporters of a lawmaker who defected to a new party on Thursday, a Dhaka medical college hospital official said.
Police and witnesses told AFP thousands of activists from the BNP and the opposition Awami League also staged demonstrations in the capital Dhaka late on Friday, lighting small fires and brandishing wooden sticks. A bus was also torched.
Dhaka police chief Mizanur Rahman said 7,000 officers were deployed and that around 15,000 would be on stand-by on Saturday to prevent any violence.
"There were some violent clashes between the supporters of the BNP and the Awami League. But the situation is under our control," he said.
At least five people were injured when Awami League members allegedly attacked and torched a Dhaka BNP leader's office on Friday, sub-inspector Musibur Rahman said.
The private UNB news agency said clashes also broke out in different district towns across the country with dozens injured. The private Channel I television network said at least 20 were injured in Dhaka.
Police earlier banned the use of sticks after the Awami League told its supporters to make hundreds of thousands of poles and oars, a reference to the party's boat emblem, in preparation for street protests.
Its mandate runs out at midnight local time (1800 GMT Friday), although the government has 15 days to transfer power to an interim administration that will oversee elections in January.
Analysts and politicians said they feared the hand over would be marred by violence.
The opposition had said it would "paralyse" the country with an indefinite transport blockade of the capital Dhaka from the moment the caretaker body takes power.
The opposition alliance has objected to the appointment of former Supreme Court justice K.M Hasan as the head of the caretaker administration.
It accuses Hasan, a former BNP official, of being pro-government and says he should be replaced as the head of the body, which is designed to ensure that ruling parties do not rig polls.

Copyright AFP (Agence France-Presse), 2006