The statement added that two Islamic state fighters shot the guards of the mosque dead, broke in and blew themselves up between two groups of worshippers, one of which consisted of around 300 people.
It was the second week in a row that militants bombed Friday prayers and killed dozens of worshippers.
HELD TO ACCOUNT
The U.N. mission in Afghanistan said those responsible should be held to account.
Taliban special forces arrived to secure the site and an appeal went out to residents to donate blood for the wounded.
Taliban spokesman Zabihullah Mujahid said security forces had been ordered to capture the perpetrators and bring them to justice under Islamic law.
The blast, coming so soon after the Kunduz attack, underlined uncertainty over security in Afghanistan as the Taliban grapple with an escalating economic and humanitarian crisis that threatens millions with hunger.
The local affiliate of Islamic State, known as Islamic State Khorasan after an ancient name for the region, has stepped up attacks following the Taliban victory over the Western-backed government in Kabul.
Taliban officials have played down the threat from Islamic State, and dismissed suggestions they may accept US help to fight the group.
But the repeated attacks have caused increasing unease from countries in the region about the potential for militant violence to spill across Afghanistan's borders.
That the Shi'ite minority has again been targeted may also inflame tensions among ethnic and sectarian groups in the country, which is largely Sunni Muslim but ethnically diverse.
Most Shi'ites in Afghanistan belong to the Hazara ethnic group of Persian speakers, who have complained of persecution under the mainly Pashtun-speaking Taliban in the past.
