AQIM’s statement – carried by monitoring group SITE – said Ouagadougou’s four-star Splendid Hotel was a “den of global espionage”.
The statement has not been independently verified.
AQIM is based in the Sahara Desert between Mali, Niger and Algeria and has attacked West African countries.
This was its first attack on Burkina Faso.
In its statement, AQIM published photographs of three men wearing military uniforms and holding guns, naming them as Battar al-Ansari, Abu Muhammad al-Buqali al-Ansari and Ahmed al-Fulani al-Ansari.
These names appear to be noms de guerre and no further details were released.
It described the hotel as “one of the most dangerous dens of global espionage in the west of the African continent”.
AQIM earlier said the gunmen were from jihadist group al-Murabitoun of Algeria’s militant Mokhtar Belmokhtar.
At least three attackers died in the assaults, officials say.
However, some witnesses were quoted as saying that they saw more than three gunmen.
Neighbouring countries have pledged to help Burkina Faso after the attack.
“We’re not going to just sit on our hands. We will react and respond,” said Benin President Thomas Boni Yayi on behalf of the West African body, Ecowas.
“The question is: ‘Whose turn is it next?” he asked, reports the AFP news agency.
Earlier, Mali, which borders Burkina Faso and experienced a similar attack last November, vowed to help the government in Ouagadougou tackle Islamist militants.
“We have decided to go forward together, hand in hand, to fight against terrorism and jihadism,” Prime Minister Modibo Keita said during a visit to Burkina Faso.
Forensic experts from France and Burkina Faso were on Monday sifting through the wreckage of the hotel for clues about the attack.
Militants attacked the Splendid Hotel and the nearby Cappuccino cafe on Friday evening, setting off several explosions.
About half of those killed were foreigners, including citizens of Canada, Ukraine, France, Portugal, Switzerland, the Netherlands and the US.
French special forces helped bring the siege to an end.