More than 25 firefighters from across Edinburgh were involved in tackling the blaze in Dumbiedykes Road early yesterday morning.
The Scottish Fire and Rescue Service (SFRS) said it received a number of 999 calls alerting them to the blaze in the block's stairway at around 4am.
All the occupants of the block had made their way to safety before the fire service got to the scene.
When they arrived at the incident, crews found the fire was well-developed on the third and fourth floors of the building. Due to the severity of the blaze, extra resources were requested by the incident commander.
Firefighters wearing breathing equipment brought the flames under control and then searched the building to make sure no-one was inside and that the blaze had not spread further.
No-one was injured but some residents were treated for minor smoke inhalation at the scene.
An investigation into the cause of the blaze is under way.
Police Scotland said they are appealing for witnesses following the "suspicious" fire.
Chief Inspector Richard Horan said: "We are investigating the cause of the fire and are keen to hear from anyone who was in or around Dumbiedykes Road in the early hours of Wednesday morning.
"You can call our non-emergency 101 number, or make an anonymous report via the charity Crimestoppers on 0800 555 111."