Rescue workers made their way through the blaze to save two people who had fled up to the roof.
Nils Kristiansen, the duty fire chief, said: "When we got there, there were open flames in the stairwell, but the most dramatic aspect was that people were hanging from the windows facing the street from the entire building."
The six people who died had all been killed while they tried to flee the building, he said. "They were discovered in the hallway and not inside their apartments."
In all, 33 people were rescued.
Oslo police said they were investigating whether the blaze was the result of arson.
Saturday's fire came just over a month after seven Polish workers died when the building they were housed in burned down in Drammen. That fire was the deadliest in Norway in more than 20 years.
Knut Storberget, Norway's justice minister, called for a nationwide review of fire safety.
"In general, we have a problem with fire prevention in apartment buildings and the government feels there is a lot to be done," he told NRK, the Norwegian public broadcaster.