A group calling itself the "Armed Islamic Movement for Al-Qaeda, the Falluja Branch" said in the tape that it, and not followers of ousted Iraqi President Saddam Hussein, had been behind the attacks.
The group's name has not been heard of before.
The tape was aired by Dubai-based al-Arabiya television. It also broadcast a photograph of a white-bearded man wearing a turban.
"I swear by God no one from his (Saddam Hussein) followers carried out any jihad operations like he claims," said the voice on the tape.
"They (attacks) are a result of our brothers in jihad," said the man.
The tape also warned of a new anti-US attack in the days to come which would "divide the back of America completely".
Al-Qaeda's claim of responsibility to attacks in Iraq came after US occupation administrator Paul Bremer blamed "bitter-enders, foreign terrorists, extreme Islamists influenced by Iran and bands of criminals" for the attacks.
Writing in the New York Times opinion section on Sunday, Bremer said Iraq's progress in establishing security, prosperity and democracy was being "sabotaged".
Bremer said the attacks against US troops and what he called "brave Iraqis working with us," were targeting the "successes in the renewal of this nation."
Bremer said his administration was "braced for an increase in terrorism by non-Iraqis."
In a Baghdad suburb, meanwhile, an Iraqi was killed when a bomb exploded near a police station.
The station in Maysaloun is frequently used by US occupation troops.
Sunday's blast was not the first that targeted the police. Last month, seven trainee police officers were killed in an explosion outside their station in Ramadi.
Members of the new US-backed Iraqi police force have been attacked in apparent retaliation for cooperating with the occupying powers.
Since the US declared major combat over in Iraq on 1 May, 77 US soldiers have been killed in the country - 31 of them in resistance operations.
Defence Secretary Donald Rumsfeld tried on Sunday to prepare the public to expect higher figures.
"I'm afraid we are going to have to expect this go on, and there is even speculation that during the month of July, which is an annivesary for a lot of Baathist events, we could see an increase in the number of attacks," Rumsfeld said.