Civil defence teams rescued more than 45 people who became trapped when the fire broke out late on Wednesday in the city of Kut, the interior ministry said.
Others are still missing, according to the state-run Iraqi News Agency.
Photographs and videos on local media showed the Corniche Hypermarket Mall, a five-storey shopping centre that had opened only a week earlier, fully engulfed in flames.
The interior ministry said 61 people died in the fire, most of them from suffocation.
Among the dead were 14 charred bodies that remained unidentified, it said.
Provincial Governor Mohammed al-Mayyeh declared three days of mourning.
He said the cause of the fire was under investigation but legal cases were filed against the building owner and shopping centre owner.
He did not specify what the charges were.
"We assure the families of the innocent victims that we will not be lenient with those who were directly or indirectly responsible for this incident," al-Mayyeh said.
The results of the preliminary investigation would be released within 48 hours, he said.
Iraqi Prime Minister Mohammad Shia al-Sudani said in a statement that he had directed the interior minister to go to the site of the fire to investigate and take measures to prevent a recurrence.
Poor building standards have often contributed to tragic fires in Iraq.
In July 2021, a blaze at a hospital in the Iraqi city of Nasiriyah that killed between 60 to 92 people was determined to have been fuelled by highly flammable, low-cost type of "sandwich panel" cladding that is illegal in Iraq.
In 2023, more than 100 people died in a fire at a wedding hall in the predominantly Christian area of Hamdaniya in Nineveh province after the ceiling panels above a pyrotechnic machine burst into flames.
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