Lance Naik (Corporal) Hanamanthappa Koppad was tapped under 8m of snow at a height of nearly 6,000m along with nine other soldiers who all died. Their bodies have now been recovered.
The critically ill soldier has been airlifted to a hospital in Delhi.
"We hope the miracle continues. Pray with us," an army statement said.
The army added that "he has been placed on a ventilator to protect his airway and lungs in view of his comatose state".
"He is expected to have a stormy course in the next 24 to 48 hours due to the complications caused by re-warming and establishment of blood flow to the cold parts of the body," the army said.
The avalanche hit a military post on the northern side of the glacier.
Senior military officials said at the time there was little chance of finding any of the soldiers alive after the incident last Wednesday.
Siachen is patrolled by troops from both India and Pakistan, who dispute the region's sovereignty.
It is known as the world's highest battlefield. Four Indian soldiers were killed by an avalanche in the same area last month.
The soldiers were on duty at an army post on the glacier at an altitude of 5,900m (19,350ft) when the avalanche struck.
Specialist army and air force teams immediately began searching for the missing soldiers close to the Line of Control that divides Indian and Pakistani-administered Kashmir.
The chances of any soldiers being found alive were so slim that Prime Minister Narendra Modi even offered condolences in a message on Twitter last week.
Avalanches and landslides are commonplace in the area during winter where temperatures can drop to -60C.
More soldiers have died from harsh weather on the glacier than in combat since India seized control of it in 1984. Soldiers have been deployed at heights of up to 6,700m (22,000ft) above sea level.
The neighbours have failed to demilitarise the Siachen glacier despite several rounds of peace talks.
An Indian soldier who was buried in an avalanche that struck the Siachen glacier in Indian-administered Kashmir six days ago has been found alive.