Nato said seven of its soldiers were killed in the crash.
"The mujahideen of the Islamic state downed the Chinook helicopter that belonged to the Nato occupiers yesterday evening using a new anti-aircraft missile," the group said in a statement posted on a website used by militant groups.
The Taliban said 60 "enemy soldiers" were killed in the crash in Helmand province in southern Afghanistan. A Chinook can carry 28 combat-equipped troops, a US military official said.
Hours after the helicopter went down, the Taliban ambushed a police convoy on a main road in southern Zabul province and killed 16 policemen, the interior ministry said.
Since a winter lull, the Taliban have launched a series of suicide attacks, roadside bombs and ambushes against foreign and Afghan forces.
In an apparent response to the Chinook crash, a joint Nato and Afghan raid supported by air power killed and wounded dozens of Taliban in Sangin district, close to Kajaki where the twin-rotor Nato helicopter went down on Wednesday night, the Afghan defence ministry said.
Nato said the fighting was part of a broader offensive called "Operation Lastay Kulang," from the local Pashto language for "axe handle," meant to extend the Afghan government's control into the north of Helmand province.
Officials said 2,000 foreign and Afghan troops, including 1,000 British soldiers and elements of the elite US 82nd Airborne Division, were involved in a push to loosen the Taliban's grip on the Sangin Valley.
Five of the soldiers killed in the crash were believed to be Americans. The other two were British and Canadian, military officials in London and Ottawa said.
The Canadian was identified as Master Corporal Darrell Priede, a military cameraman serving with the army.
At least 55 US soldiers have been killed in five previous Chinook crashes in Afghanistan in the past two years.
Helmand, a long-time bastion of the Taliban, is the leading drug-producing region of Afghanistan, the world's main heroin supplier. It has been the scene of a series of bloody fights between Taliban and Western forces in recent months.
Taliban leaders have threatened in recent weeks to step up attacks on foreign troops and said they have trained hundreds of suicide bombers.
The guerrillas are active largely in southern and eastern areas of Afghanistan, where more than 5,000 people have been killed in the past 16 months.
The violence is rising despite the presence of nearly 50,000 foreign troops led by Nato and the US military as well as about 100,000 Afghan security forces.
