"A very tragic night at the beginning of a holiday weekend", City Fire Chief John Fruetel said at a press conference, according to the Associated Press.
The department reported that crews evacuated a number occupants out of the building through the stairwell.
One person who could not use the stairs to escape had to be helped out from the 13th-floor elevator, fire officials said.
The building is part of a complex known as The Cedars. The name of the fifth victim was not released, but authorities said all the people who died were adults.
"She said, 'I open the door and I smelled, and I hear the noise and I run, '" Shire said.
A displaced resident looks out a window after a deadly fire at a high-rise apartment building on November 27, 2019, in Minneapolis. "Crews did an outstanding job, remembering this is a 24-story building".
Warsame, who is Somali-American, said he knew that one victim was a Somali grandmother.
A spokesperson for the Minneapolis Public Housing Authority told NBC affiliate KARE that the 14th floor was closed off, but other building residents could return home within the day.
Floors above the 14th were being checked for habitability. Horwich said residents on the lower floors would likely not be displaced.
Given its age, the building is not required by city codes to have a sprinkler system.
City officials said public housing inspections are handled by federal agencies, and that the city inspected the building only to respond to specific complaints.