The other batch that also left on Thursday will travel through the old Lipulekh Pass route.
"When I flagged off the yatra last year, I promised we will try to open Nathu La Pass from next year for the pilgrims. I am happy to fulfil the promise," Sushma Swaraj told the pilgrims.
Kailash Mansarovar is believed to be the seat of Hindu god Shiva. Pilgrims travel at high altitudes of 19,500 feet. The traditional trekking route has been inhospitable and rugged.
Hundreds go on the pilgrimage every year, a part of the journey overseen by Chinese authorities.
Eighteen batches of 60 people each will pass through the old route whereas five batches of 50 people each will use the new route to reach Kailash Mansarovar, the minister said.
The new motorable route via Nathu La on the India-China border from east Sikkim to Shigatse, the second biggest city in Tibet, was added following a MoU during Prime Minister Narendra Modi`s visit to Beijing in 2014.
"We have two advantages with the new route. The old and the disabled can undertake the yatra by car till Mansarovar. We can also send more people," Sushma Swaraj said.
The new route will enable the pilgrims to drive all along, travelling first from Gangtok in Sikkim to Shigatse in Tibet from where they can take vans and buses on the existing road to Mansarovar and Kailash.
The minister said the government had introduced a voice response system to ensure a safe journey for the pilgrims.
"The voice response system will help us communicate with the pilgrims effectively and to address their problems," she said.
Wishing the pilgrims a safe journey, Sushma Swaraj said that only the chosen ones get a chance to undertake the pilgrimage.
"Consider yourself lucky for getting this opportunity.... So many people have applied but your luck worked.... It is a direct blessing from Bhagwan Shankar (Lord Shiva)," she remarked.
External affairs ministry spokesperson Vikas Swarup tweeted: "A holy pilgrimage begins."
While pilgrimage through the old route via Lipulekh Pass will cost each pilgrim Rs.1.5 lakh, the new route will cost Rs.1.7 lakh.