Authorities responded to the blaze at the Granville Towers in the 1400 block of Crescent Heights Boulevard just before 10 a.m. The fire was on the third floor of the seven-story building.
Three people had minor injuries, but declined being hospitalized. One of Greene's two dogs died in the fire, according to L.A. County Fire.
Anna Lobell said she lives right below the unit destroyed in the fire. "It was really chaotic. There was just a lot of noise, and people were screaming and saying, 'Get out of the building,'" Lobell said. "It was scary."
The building was evacuated before firefighters arrived.
"I grabbed the dogs, grabbed a robe and just bounced," said Laney Ota, a resident.
It took firefighters from L.A. County, L.A. City and Beverly Hills around 15 minutes to put out the fire.
"The fire was limited to the room of origin, primarily content," said L.A. County Fire Inspector Tony Imbrenda.
According to its website, the Granville Towers was built in 1930 and has been the home to a number of Hollywood celebrities, including Marilyn Monroe and Rock Hudson.
The cause of the fire is unknown at this time.
NOTE: It was earlier reported that two people were taken to a local hospital, but fire officials now say no one was transported.