The strike was called to protest the deaths of three people in clashes between police and demonstrators protesting the life sentence given by a Bangladeshi tribunal to Abdul Kader Mullah, chief of the Jamat-e-Islami group, for his alleged role in crimes during the country's 1971 independence war.
Members of the so-called Shahbagh movement want Mullah executed.
The Bangladesh Daily Star said no incidents were reported as the strike began Monday. The report said public transportation systems were operating in the capital Dhaka, but there were fewer private vehicles on the roads.
Security remained tight in the capital as supporters of the Shahbagh movement urged the public to ignore the strike and go about their daily routine.
Thousands of Shahbagh supporters have been holding demonstrations ever since the sentencing of Mullah, demanding his execution as well as a ban the pro-Jamaat blog and the immediate arrests of those responsible for last week's killing of Ahmed Rajib Haidar, a key movement supporter. They also want death penalty for all war criminals and no grant of amnesty.
The Star reported many had heeded the call of Shahbagh supporters to defy the Jamaat's strike call.
As the Shahbagh demonstrations continued, Bangladesh's parliament amended a law to allow the state to appeal the tribunal's life sentence of Mullah, which was cheered by tens of thousands of the demonstrators, the BBC reported.
Bangladesh, formerly the eastern wing of Pakistan, became an independent republic in 1971 after a nine-month war against the Pakistani army. Some estimates say as many as 3 million people died and tens of thousands of women were raped and attacked during the war.
The Bangladeshi tribunal was set up in 2010 to try some of the local groups who allegedly opposed the freedom war and collaborated with the Pakistani army in aiding the killings, rapes and other atrocities.
The Jamaat claims the latest efforts are part of a political vendetta by the government.
The BBC reported Mullah and seven others accused are Jamaat party leaders, while the other two are members of the opposition Bangladesh Nationalist Party, a rival of Prime Minister Sheikh Hasina and her ruling party.
The tribunal does not conform with international standards, human rights groups have said.
Mulah's sentencing was the second by the tribunal. On Jan. 21 Jamaat member Abul Kalam Azad was sentenced, in absentia, to death.