One protester was killed last night in the northern town of Chirirbandar in protests ahead of Tuesday's shutdown.
A special court is trying senior leaders of Jamaat-e-Islami for allegedly carrying out atrocities during the  1971 independence war.
The party says the charges are politically motivated.
Protesters are out on the streets across the country in support of Jamaat-e-Islami's strike and there are reports of sporadic clashes.
In some areas demonstrators have set vehicles and tyres on fire as well as  pelting police vehicles with stones.
Schools and businesses in many parts of the country have been shut.  Thousands of police officers have been deployed.
"We have sufficient police force now on the streets... Many vehicles are moving on the street and we are giving them protection and also activists are suddenly attacking, coming out on the street and trying to disturb, we are also looking into that," one  police officer in Dhaka told the Reuters news agency.
The strike came a day after the authorities refused permission for the party to hold a rally. The main opposition Bangladesh Nationalist Party has also extended its support for the strike.
The BBC's Ethirajan Anbarasan in Dhaka says that many people are staying indoors fearing violence, with most private vehicles off the roads.
As some of the cases at the special tribunal into war crimes have reached their final stages, there are concerns that Jamaat-e-Islami may intensify its protest.
Analysts believe this could be the beginning of a series of general strikes, with Jamaat-e-Islami determined to show its opposition the war crimes trials and the BNP opposition also threatening to call general strikes over a dispute on restoring the country's caretaker administration for parliamentary polls.
Our correspondent says that most people in Bangladesh are fed up of such strike action, and many fear the country may be in for a prolonged period of instability.
Violence has flared in Bangladesh as the country's largest Islamist party enforces a general strike to demand the release of its leaders from prison.