Chicago fire officials initially said the blaze was caused by a "pot of meat" in a seventh-floor apartment unit in the 400-block of West Wrightwood Avenue.
Residents of the building said people began banging on their doors to alert them to the fire around 5 p.m.
One woman said she was sleeping at the time, and not even the sirens woke her up.
"I was taking a nap, so I didn't even hear the sirens. I just heard a lot of glass breaking and they were banging on everyone's door," Shayna Soloway said. "They were like, 'You have to get out. You have to get out.' And I was like, 'I'm not leaving without my cat!"
Once emergency crews arrived, police and property managers quickly moved door to door on the lower six floors.
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"The building property manager came and just started pounding on my door, yelling that there is a fire in the building and we all needed to get out," said Kyle Bentz, a resident. "I grabbed my coat, put my shoes on and got out of there."
"The first that went through my head was like 'Get all the pets.' Get anything that's living, alive, that I can save and rescue and try to evacuate," said Convontae Patton, a resident.
A 31-year-old man was found on the ground near the building, and Chicago police said he "appears to have jumped" from the seventh floor. He was transported to Illinois Masonic Medical Center in serious-to-critical condition, where he later died.
The Cook County Medical Examiner's Office identified him as Demarlo Branch.
Four responding officers suffered minor injuries due to smoke inhalation, and were transported to the same hospital for evaluation, fire officials said.
No other residents were displaced in the fire, which was contained to one unit.
Everyone was expected to be allowed back into their homes Monday.
The Sun-Times Media Wire contributed to this report.