The clashes took place as US and Iraqi forces combed the devastated streets of the former rebel enclave for the last remaining militants after a battle that was launched one week ago.
The latest death takes to 39 the number of US soldiers who have died since the assault on Fallujah was launched last Monday, according to military figures.
At least five Iraqi troops and more than 1,200 rebels have also died.
The casualties are virtually impossible to independently verify.