The explosives were packed in a truck that detonated in the Souk Jameela neighborhood in the eastern part of Baghdad.
In addition to the 36 dead, Iraq's Interior Ministry reported 75 wounded from the blast.
The top U.N. official in Iraq , Gyorgy Busztin, gave a higher death toll of 45 people.
"These figures tragically are expected to rise," Busztin said in a news release. "I utterly condemn this heinous and cowardly attack, targeting innocent civilians, many children included, in a public marketplace."
A statement circulated online by ISIS supporters claimed responsibility for the blast in what they described as a "blessed operation" targeting Shiite militia and members of the Popular Mobilization Units in one of their strongholds in Baghdad's Sadr City area.
The Popular Mobilization Units, predominantly Shiite militias allied with the government, have been deployed alongside Iraqi forces to try to push back against ISIS, which consists of Sunni Muslim extremists.
Such attacks with large death tolls occur with sad regularity in and around Baghdad.
Earlier this week, a suicide car bomb killed 61 people in the city of Baquba
, 40 miles (65 kilometers) north of Baghdad. ISIS, which has targeted areas not far from the Iraqi capital, claimed responsibility for the Baquba bombing. That blast hit a mostly ethnic Shiite area.
ISIS has also struck Shiite areas closer to Baghdad. Last month, a truck laden with explosives killed at least 120 people in an outdoor market
in Khan Bani Saad, a town some 32 kilometers (20 miles) north of Baghdad.
U.S. State Department spokesman John Kirby, who until recently had that same role at the Pentagon, said the Baghdad and other recent attacks "show once again the utter disregard (ISIS) has for innocent civilians, including women and children."
ISIS has employed this approach with success while taking over vast swaths of Iraq and neighboring Syria, saying its faith justifies large-scale killings, taking slaves and raping girls and older women who are "nonbelievers
Busztin predicted this approach won't work.
"I have no doubt in my mind," he said, "the terrorists will fail."