The blaze started at around 4am on Thursday in Green Court, with residents evacuated by emergency services called to a “serious” fire.
Bedfordshire Police said one person was confirmed to have died at the scene, with a “small number” of people taken to hospital.
All occupants have been accounted for, the force said, and the fire has now been extinguished.
A Bedfordshire Police spokesperson said: “A small number of people including residents and police officers have been taken to hospital for treatment, and sadly one person has been confirmed to have died at the scene.
“All occupants have been accounted for.
“Work is ongoing at the site and several areas have been cordoned off as investigations continue.
“We understand this is a difficult time and people will want to know if loved ones are safe.
“We would ask people not to speculate or share images of the incident as it may cause distress to those involved. We will provide regular updates as soon as we have them.”
The force said it is working with others, including Luton Council and Bedfordshire Fire and Rescue Service, to look after residents who have been evacuated while the building is made safe.
The fire service said in a tweet: “Bedfordshire Fire and Rescue Service are currently at the scene of a serious flat fire in Luton. 10 appliances are in attendance. The fire has now been fully extinguished and crews are working to ensure there are no continuing hotspots.”
It added the fire investigation team is at the scene.