Fire crews were called to the blaze in Sumburgh Street, Cran- hill, at around 6am.
The alarm was raised after the fire broke out at the complex managed by Blackwood Homes.
Strathclyde Fire & Rescue said the blaze started in the ground floor kitchen area.
Firefighters wearing breathing apparatus searched a ground floor flat while their colleagues battled the flames.
As crews fought the "well developed" fire a resident was found in the building. Sadly, paramedics pronounced the person dead at the scene.
At around noon today police were still at the scene of the tragedy where the windows were covered with tarpaulin.
Officers could be seen going in and out the flat today.
A Scottish Gas Networks van was also parked outside the property.
It is not yet clear what caused the blaze.
A joint investigation has now been launched by Strathclyde Fire & Rescue and Strathclyde Police.
A Strathclyde Fire & Rescue spokesman, said: "The crews were faced with a well-developed kitchen fire and the crews worked tirelessly in difficult conditions to find the individual and prevent the spread of fire to other parts of the house."
A police spokeswoman confirmed officers were called to the blaze at around 6.10am today.
The force did not release any details of the householder who died.
A spokeswoman for Blackwood Homes, said: "Blackwood is deeply saddened at the news that one of our tenants has died in a fire at their home.
"Initial reports indicate the fire was located in the kitchen area and was discovered at 6am this morning.
"Our thoughts are with the tenant's family at this very sad time.
"Staff are currently on site working with emergency services.
"We have no firm information about the cause of the fire at present.
"However, we can confirm that there are no suspicious circumstances and this was a tragic accident."
Fire officers stressed the importance of fitting smoke alarms
The spokesman added: "We need to accept that fire can strike anywhere and take action to protect ourselves, our families, friends and neighbours.
"It is absolutely vital that every home has a working smoke alarm."