Taliban provincial police said four fighters were killed and 10 arrested in at least four districts in Kandahar province.
The Taliban have launched a crackdown on suspected Daesh hideouts in southern Afghanistan following an increase in bloody attacks by the group in recent weeks.
The operation against Daesh-Khorasan, started around midnight on Monday in at least four districts of Kandahar province and continued through the morning, according to Taliban provincial police chief Abdul Ghafar Mohammadi.
"So far, four Daesh fighters have been killed and ten arrested... one of them blew himself up inside a house," he said.
A member of the Taliban intelligence agency who declined to be named said at least three civilians have been killed in the operation.
Daesh also claimed responsibility for a bomb attack on Sunday that destroyed a minibus in Kabul killing a well-known local journalist and up to two others.
The group claimed it had killed or injured "20 Shiite apostates" in the incident.
Daesh after Taliban takeover
In the three months since the Taliban came to power, Daesh has been active in Jalalabad, Kunduz, Kandahar and Kabul.
Earlier this month Daesh fighters raided the city's National Military Hospital, killing at least 19 people and injuring more than 50 others.
The group has also claimed several attacks in the city of Jalalabad, the capital of eastern Nangarhar province and a hotbed of Daesh activity.
Last month the group claimed responsibility for a suicide bomb attack on a Shia mosque in Kandahar that killed at least 60 people and injured scores more.
That attack came a week after another deadly mosque blast claimed by Dash in northern Kunduz province killed more than 60 people.