The terrorist group made the announcement in an online statement.
The militants transported a truck bomb in Sadr City district to carry out the operation at a wholesale marketplace in the Jamila district, said the statement the authenticity of which could not be independently verified.
A police source said that 53 people died and over 100 others were injured as a refrigerator truck packed with explosives and carefully camouflaged with vegetables and fruits, detonated at the Jamila wholesale vegetable market in the Shiite bastion of Sadr City district in eastern Baghdad.
Several people went missing by virtue of the massive explosion, which destroyed numerous shops and stalls and set alight a few civilian cars, the source said, Xinhua News Agency reported on Thursday.
Iraqi security forces cordoned off the area to secure the scene, while ambulances and civilian vehicles evacuated the victims to several hospitals and medical centers.
ISIL frequently targets areas where crowds gather, including markets, cafes and mosques across Iraq.
The security situation in Iraq has drastically deteriorated since June 2014, when bloody clashes broke out between security forces and ISIL militants.
The United Nations Assistance Mission for Iraq (UNAMI) previously reported that an estimated 14,136 civilians died while 29,189 others were injured in the Iraqi armed conflict since January 1, 2014, up to the end of April 2015.