Haji Ajab Shah was killed on Tuesday at the police headquarters of the city, the capital of Nangarhar province.
"His body is in the morgue," said a doctor at the local hospital where Shah was taken. The size of the bomb or how it was detonated was not immediately known.
An aide to Shah was also wounded in the blast, which occurred at the start of the official day, police sources said.
Jalalabad, which lies 130 km east of Kabul, has been hit by a series of blasts targeting government offices and Western aid agencies, but most of them have not caused serious casualties.
Spate of attacks
The attack was the latest sign of rising violence, mainly in the Afghan south and east, ahead of Afghanistan's landmark elections planned for September.
Authorities have blamed members of the ousted Taliban regime and Islamic fighters for the spate of attacks in which more than 700 people have been killed since August.
The Taliban, overthrown in a US-led war in late 2001, have vowed to disrupt elections, which President Hamid Karzai is expected to contest and win.
Violence has also been linked to tribal rivalry and the illicit drug trade, which is rampant in parts of the south and east close to the border with Pakistan.