Police say a young man selling tea detonated the bomb hidden in a flask when they were about to frisk him outside an administrative building in Gazipur town.
Gazipur, close to the capital Dhaka, is one of two towns that were hit by suicide bomb attacks earlier this week. Those attacks killed 10 people and wounded dozens.
Both those bombs also targeted courthouses, and prompted the Bangladesh Supreme Court Bar Association to call for a nationwide strike, which paralyzed the country on Thursday.
Schools, courts and offices shut down, and lawyers held protest marches in several towns demanding greater security.
The government has blamed a banned Islamic militant group called the Jamaat-ul-Mujahedin for the attacks. The group wants strict Islamic law imposed in the country. It has been accused of conducting a string of earlier bomb blasts that targeted the judiciary, politicians and government buildings.
A security analyst, retired Brigadier General Shakawat Hossain, says the rising tide of violence has created a climate of fear in a country regarded as a moderate Muslim nation.
"It has come as a surprise, not only to society, but even to law enforcing agencies who are not designed and trained to treat such kind of onslaught. Whole society has been taken aback," he said.
Police say Tuesday's bomb attacks were the first suicide bombings in the country and demonstrated that the militants are becoming more sophisticated. They say the Jammat-ul-Mujahideen may have trained hundreds of bombers, and police have launched a massive manhunt to track down the militants.
The attacks have triggered calls for greater political unity in the country, where the two main parties have been bitter rivals for years.
Mr. Hossain says Bangladesh will require greater political resolve to tackle the militants.
"Pressure is building up from civil society. A huge debate is going on this particular issue that there has to be a political consensus to address this issue," he said.
Prime Minister Khaleda Zia has vowed tough action against the bombers, and says her government will do "everything and anything" needed to stop them.