"We tell the enemies that these [operations] are proof that God is giving us victory," the group said in a statement posted on a website used by Sunni groups.
"Hundreds of people saw the aircraft, their voices rising in chants of Allahu akbar [God is greatest]," the statement said.
Earlier, witnesses said the helicopter had been shot down in a field in the Sheik Amir area northwest of Baghdad
, sending smoke rising from the scene, in a Sunni-dominated area between the Taji air base 20km north of Baghdad
and Garma, 30km to the west of the capital.
Rash of crashes
A CH-46 has two crew members and can carry up to 55 passengers. They are used widely to transport troops and cargo around Iraq.
The reports came five days after a US army helicopter crashed in a hail of gunfire north of Baghdad, police and witnesses said - the fourth helicopter lost in Iraq in a two-week span.
US command said two crew members were killed in that crash.
The US military said on Sunday it was adjusting its tactics in Iraq after four helicopters were shot down in recent weeks.
Those helicopters were shot down in four separate incidents in which 21 US servicemen and private security contractors were killed.