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President Ashraf Ghani blamed Islamic State militants, without giving further detail. If true, it would be the first such major attack carried out by the group in Afghanistan, marking a significant step in its expansion into South Asia.
Up until now militants claiming allegiance to Islamic State in Afghanistan have been widely identified as former Taliban fighters disillusioned with their leadership. The Taliban itself condemned the attack as "evil".
The explosion smashed windows and sent debris flying across a tree-lined street, filling the air with smoke and dust.
"It was a suicide attack," police chief Fazel Ahmad Sherzad told a news conference. He added that officials were investigating witness reports of a second explosion after people had rushed to the area to help the wounded.
Police said a later blast that shook Jalalabad was a controlled detonation after experts discovered a further bomb close to the scene of the initial explosion.
Local media said a former spokesman for the Pakistani Taliban had claimed responsibility on behalf of the Islamic State in both Pakistan and Afghanistan.
The spokesman could not be reached and his connection to the group could not be verified by Reuters.
Islamist militants of various hues already hold sway across restive and impoverished areas of South Asia, but Islamic State has started to draw support from younger fighters in the region, impressed by its rapid capture of territory in Syria and Iraq.
President Ghani visited Washington last month and warned that Islamic State posed a "terrible threat" to his country.
Taliban insurgents denied responsibility and did not comment on the alleged Islamic State link. The militants, who were ousted from power by a U.S.-led invasion in 2001, rarely claim attacks that kill large groups of civilians, saying they target foreigners or the Afghan military and government.
"It was an evil act. We strongly condemn it," the Taliban spokesman Zabihullah Mujahid told Reuters.
Elsewhere, in the eastern province of Ghazni, the Taliban announced the execution of three men accused of murdering a couple during a robbery, saying they had been tried by an Islamic court.
The men were shot dead in front of a crowd by Taliban fighters, according to a Reuters witness. A video of the killings showed the men were made to sit with their eyes blindfolded and their hands tied at the time of their execution.
"(They) killed a female doctor and her husband ... inside their house and then they took all their assets, jewelry and cash," the Taliban said in a statement.
After the initial shots were fired, one fighter continued to shoot at the bodies while the crowd of villagers cheered.
This is the first year Afghan forces are facing the Taliban with very limited international support on the ground, although help with intelligence and special operations will continue through 2016.
On Saturday, parliament approved the nomination of 16 ministers, leaving only the position of defence minister vacant because the president and his coalition have been unable to agree on a candidate.
The vacancy has frustrated military officials who say the army has been left rudderless in the face of escalating violence by the Taliban. NATO, which at its peak had 130,000 soldiers in Afghanistan, has only a small contingent of around 12,000 troops left and most are involved in training.