The blaze at a flat at Priorwood Court in Melrose was reported to fire crews just before 04;30.
Firefighters used a line of hosereel and six breathing apparatus to tackle the flames.
The fire is understood to have been contained to one flat only.
A spokeswoman for the Scottish Fire and Rescue Service said: "One elderly lady was rescued from the flat and given oxygen by the fire service. She has been taken to hospital by ambulance."
An elderly woman has been taken to hospital following a fire at sheltered housing complex in the Scottish Borders.