The Taliban's chief spokesman Zabihullah Mujahid said fighters carried out the operation in the north of Kabul on Sunday evening. "As a result of the operation, which was very decisive and successful, the ISIS center was completely destroyed and all the ISIS members in it were killed," Mujahid said on Twitter.
Earlier, local media reported heavy clashes in the area, and residents confirmed they had heard explosions and gunfire during the night.
There was no confirmation that the operation was directly connected with Sunday's blast, which appeared to be the most severe attack in the Afghan capital since the withdrawal of U.S. forces at the end of August.
The Taliban took control of Kabul in mid-August and have set up an interim Afghan government, but still face attacks from the regional branch of ISIS, known as ISIS-Khorasan.
ISIS has claimed responsibility for bomb attacks in the eastern city of Jalalabad as well for a suicide attack that killed 13 U.S. troops and scores of Afghan civilians who had crowded outside the Kabul airport gates, desperate to secure seats on evacuation flights.