Three women and one man, all in their late teens to early twenties, suffered stab wounds to their upper body and face injuries after a fallout in Muirhouse View.
All four were taken to the Royal Infirmary of Edinburgh but their injuries were not described as life threatening.
The fight happened at around 5am leading to a number of police being called as well as three ambulances and a paramedic response car.
Six people were in the flat and police confirmed that they were all known to each other.
A Lothian and Borders Police spokesman said: “Police are investigating an incident that happened at a property in Muirhouse View around 5am today, where four people suffered stab injuries.
“They were all taken to the Edinburgh Royal Infirmary for treatment to their injuries, which are not life-threatening.
“Officers are following a positive line of inquiry in relation to the incident.”