Ms Swaraj told the pilgrims to take care to not "sully" the route leading up to Kailash Mansarovar during the holy journey.
"The reverence we have for Lord Shiva, we should have similar respect for his abode. Therefore, please do not sully the road that will lead you to Lord Shiva," she told the pilgrims at the Jawaharlal Nehru Bhawan in New Delhi.
A total of 1,430 devotees will undertake the pilgrimage in 25 batches this year. Eighteen batches, each comprising around 60 pilgrims, will visit Kailash Mansoravar through the more arduous Lipulekh Pass route, while seven batches with 50 pilgrims each will take the newly-opened Nathu La route.
The yatra will continue for the next four months.
The MEA had received 4,442 applications this year compared to last year's 2,600. The pilgrims were shortlisted through a computerised draw of lots.
Ms Swaraj said the Indian government was making constant efforts and improvements to make the journey more convenient for the pilgrims.
"But it is a gradual process and we are doing a bit more every year," she said.
Sushma Swaraj, who underwent a kidney transplant in December last year, asked the pilgrims to seek blessings on her behalf as well.
"I underwent the operation but by God's grace I could recuperate within months. Since you all are going to seek blessings of Lord Shiva, I'd request you to seek some blessings for me too as I myself haven't gone to Kailash Mansarovar, ever, and am not sure if I ever will," she said.