Gunmen killed a total of 17 people in three days of violence that began when they opened fire at Charlie Hebdo last week in revenge for publication of satirical images of the Prophet.
This was the first time that a group officially claimed responsibility for the attack, led by two brothers who had visited the poor Arabian peninsula country in 2011.
"As for the blessed Battle of Paris, we... claim responsibility for this operation as vengeance for the Messenger of God," said Nasser bin Ali al-Ansi, a leader of the Yemeni branch of al Qaeda (AQAP) in the recording.
Ansi, an AQAP ideologue, said the "one who chose the target, laid the plan and financed the operation is the leadership of the organisation", without naming an individual.
He added without elaborating that the strike was carried out in "implementation" of the order of overall al Qaeda leader Ayman al-Zawahri, who has called for strikes by Muslims in the West using any means they can find.
Ansi also gave credit for the operation to slain AQAP propagandist Anwar al-Awlaki, a preacher cited by one of the gunmen in remarks to French media as a financer of the attack.
It was not clear how Awlaki, killed by a U.S. drone in 2011, had a direct link to the Paris assault, but he inspired several militants in the United States and Britain to acts of violence.
The group mocked a big solidarity rally in Paris on Sunday for the victims, saying the shock on display showed feebleness.
"Look at how they gathered, rallied and supported each other; strengthening their weakness and dressing their wounds," it said of Western leaders who attended the event.
It was not immediately possible to verify the authenticity of the recording, which carried the logo of al Qaeda's media group al-Malahem.
The purported claim of responsibility puts a fresh spotlight on a group often cited by Western officials as al Qaeda's most dangerous branch. AQAP has recently focused on fighting enemies at home such as government forces and Shi'ite rebels, but says that it still aims to carry out attacks abroad.
Two senior Yemeni sources that both Cherif and Said Kouachi, who carried out the Charlie Hebdo attack, had travelled to Yemen via Oman in 2011, met Awlaki and underwent weapons training in the eastern province of Marib.
A Marib tribal leader denied that the brothers had trained there in 2011 or that Awlaki used to be present in the province.
AQAP's Yemeni leader, Nasser al-Wuhayshi, was once a close associate of al Qaeda leader Osama bin Laden, whose father was born in Yemen, a neighbour of top oil exporter Saudi Arabia.
Yemeni President Abd-Rabbu Mansour Hadi complained on Wednesday that Yemen was subject to a politicised media campaign over the attackers' 2011 visit, state news agency Saba said.
"The person reported to have travelled to Yemen to learn in three days how to fire a pistol had been detained and under investigation for two years in France," Saba quoted Hadi as saying. Hadi wondered why such suspicious elements came to Yemen and returned home without being questioned, it said.