ISIS claimed responsibility for the attack while the Taliban denied involvement.
Dozens of unarmed Taliban militants had earlier entered the Afghan capital and other cities to celebrate Eid and the end of Ramadan.
Soldiers and militants took selfies on their smartphones and exchanged hugs.
A spokesman for the provincial governor of Nangarhar, Attaullah Khogyani, confirmed a car bomb was responsible for the blast in the town of Ghazi Aminullah Khan, on the main Torkham-Jalalabad road, and said dozens were wounded.
But he had said earlier a rocket-propelled grenade was to blame.
The Taliban announced a three-day ceasefire over the Eid holiday, which began on Friday following the end of the Ramadan fasting period.
In a message sent on WhatsApp, the Taliban said: “All the Mujahideen are directed to stop offensive operations against Afghan forces for the first three days of Eid-al-Fitr.”
However, the group sent a worrying warning to foreign forces in the region.
They said “foreign occupiers are the exception” in the ceasefire.
They added: “Our operations will continue against them, we will attack them wherever we see them.”
The Afghan government is also carrying out a ceasefire which will last until Wednesday.
President Ashraf Ghani said in an address to the nation he would extend the ceasefire with the Taliban but failed to give a time-frame.
He also asked the Taliban to extend their three-day ceasefire, which is due to end on Sunday and begin peace talks.
It was not clear if Mr Ghani was aware of the bomb attack when he made his address.
US Secretary of State Mike Pompeo echoed Mr Ghani’s address.
He said peace talks would have to include a discussion on the role of “international actors and forces”.
Mr Pompeo added: “The United States is prepared to support, facilitate, and participate in these discussions.
“The United States stands ready to work with the Afghan government, the Taliban, and all the people of Afghanistan to reach a peace agreement and political settlement that brings a permanent end to this war.”