Neighbour Richard Scott Robinson, who lives in the flat above, said that he heard her shuffling towards the door as he repeatedly banged it to get her out.
Fire-fighters arrived soon after and hauled the 79-year-old woman, who named locally as Berit Mountain, from her flat in Argyle Park Terrace.
She was rushed to the Edinburgh Royal Infirmary by ambulance crews suffering from serious injury and later transferred to the burns unit in St John’s Hospital in Livingston where she failed to recover and died.
Around 20 fire fighters from Tollcross and Liberton worked to extinguish the “ferocious” blaze that broke out around 5.40pm on Monday, managing to help rescue three others.
Mr Scott Robinson said that he was alerted to the fire when he could smell smoke and after having a thorough check of his own flat he went into the hallway.
He said: “I opened my front door and I just noticed a whisper of smoke, just like you would get off a cigarette so I phoned 999.
“The fire brigade were already on their way. After I rang the fire brigade I just went down and started banging on the door.
“There was only a small whisper of grey smoke above the door.
“I had already been banging on the door for quite a long time before I heard the footsteps. I was worried that I might make things worse by kicking the door in.
“They told me later that they found her behind the door.
“Through me banging on the door she must have come along the passage, but she must not have had any strength left.
“You would expect someone to be screaming but that didn’t happen.”
Mr Robinson added: “I had no idea of the extent of the fire until I got outside.
“It was like a bonfire with six foot high flames coming out the window.
“I was astounded.
“When I passed her door on the way up to my flat, when I looked in the floor was completely black.”
The flames shattered the window of the first floor flat and left scorch marks down the side of the building.
Mr Robinson added: “We used to bump into each other on the stairs and say hello.
“I feel similar to how I felt when my father died; in shock and a bit emotionally fragile.”
Mr Robinson said that Mrs Mountain was originally from Sweden and had lived in the flat for about 18 years.
He said that her late husband Bill was from Northeast England and they had a son who lived in America and a daughter from Edinburgh.
It was thought that Mrs Mountain used to volunteer for homeless charity The Ark and that she may have been related to a famous Swedish artist.
Mr Robinson added: “I was told that they collected her daughter and she was with her mother at the burns unit.”
Mr Robinson said that he didn’t hear a smoke alarm to alert him to that fire but that Mrs Robinson was a heavy smoker.
David Ross, 34, and Stacy Schwartz, 26, were also rescued by fire crews with a turntable ladder.
Both were treated on scene for smoke inhalation but did not need to go to hospital.
Their cat, Fudge, also had to be rescued by fire crews.
Miss Schwartz said that Mrs Mountain used to collect money for the stair cleaning.
Mr Ross said: “I don’t think I lost my head. I was quite calm.
“The bathroom was just filled with smoke that was coming through the vents.
“But I could hear sirens so I knew the fire crews were on the scene.”
David Lockhart, group manager for Lothian and Borders Fire and Rescue said: “The fire service received six 999 calls at 5.41pm to this address and were on scene at 5.45pm to find a well developed fire with flames coming out of the window.
“Six breathing apparatus, a hose reel and a turntable ladder were used.
“They rescued a 79-year-old lady who was attended to on scene and was taken to the Royal Infirmary of Edinburgh with police escorts.
“We rescued two other people from one of the upper floor flats. A forth casualty self-presented.
“All were treated by ambulance staff.
“At this moment in time fire investigation is still underway. We would like to emphasis we have seen five fire deaths across Strathclyde and Lothian and Borders recently and it brings it home that a fire in your home is a very real risk.
“It looks like the fire started in the bedroom. Neighbours have said that she was a smoker but we are not sure if that was the cause.”
A spokesman for Lothian and Borders Police said: “There was a serious fire and we are continuing our enquiries.”