Nearly 60 firefighters and eight fire engines battled the large blaze on the 12th floor of the 20-storey block in Elmira Street, Lewisham, at 4.14am.
About two hours later, the London fire brigade declared the blaze under control.
A sprinkler system had been able to suppress the fire, however residents complained about the lack of fire alarms.
Resident, Caleb Parker, 40, who lives on the 10th floor, said: “I was sleeping and someone banged on the door and I got up and didn’t know what it was.
“I heard someone else banging on the door saying, ‘fire, get out’.
“I wasn’t sure what to expect and when I saw the fire I thought it could be really bad.”
He added: “Up to now I felt safe. I don’t know that I feel unsafe, but I’m concerned, certainly on the one-year anniversary of Grenfell.”
Another resident, Alix Mabon, 25, who lives on the fourth floor of Roma Corte, was one of the first people out of the building.
She praised the bravery of three teenagers who alerted her to the fire before running through the building knocking on doors.
“I slept with the balcony door open,” the civil servant said. “I just heard people screaming and shouting and thought it was an argument on the street, but it just got louder and louder.
“When I came out three teenagers were with the concierge shouting, ‘fire, fire. Get out’.
“The first thought that came into my head was, ‘Oh my God, it’s Grenfell. We’re going to be the next Grenfell’.”
Crews from Lee Green, Deptford, New Cross, Forest Hill and Lewisham were among those sent to tackle the blaze, the London fire brigade said.
Met Police said a nearby leisure centre was used to accommodate residents who were evacuated from their flats.
No injuries were reported and the cause of the fire was being investigated.
Lewisham Council said it had helped with the evacuation, and that residents were being allowed to return to the building on Thursday morning.