The prison houses about 1,500 inmates, of which several hundred are believed to belong to the Islamic State group affiliate in Afghanistan.
The attack comes a day after the Afghan intelligence agency said a senior Islamic State group commander was killed by Afghan special forces near Jalalabad.
While the Islamic State group has seen its so-called caliphate stretching across Iraq and Syria eliminated after a yearslong campaign, the group has continued fighting in Afghanistan.
The extremists also have battled the Taliban in the country, whom the US overthrew following the 2001 American-led invasion after the Sept. 11 attacks.
The Taliban’s political spokesman, Suhail Shaheen, told The Associated Press that his group was not involved in the Jalalabad attack.
The US struck a peace deal with the Taliban in February.
A second, crucial round of negotiations between the Taliban and the political leadership in Kabul has yet to start.
Still, Washington and NATO already have begun withdrawing troops in line with the deal.
“We have a cease-fire and are not involved in any of these attacks anywhere in the country,” he said.
The Taliban declared a three-day ceasefire starting Friday for the major Muslim holiday of Eid al-Adha.