Tsering Dorjee, a Facebook friend and Sangay's admirer, wrote: "wow for the first time India did something. Thank you BJP".
"Thank you Modi-ji in the BJP for your respect to exile Tibetan government," wrote another friend Tashi Dorjee.
Sangay, the democratically elected leader and the political successor to Tibetan spiritual leader the Dalai Lama, was specially invited to attend the impressive swearing-in ceremony, his aide told IANS.
India is home to the Dalai Lama and some 100,000 Tibetan exiles. The Dalai Lama is based in this northern hill town.
"Honorable Sikyong (political leader) attended the swearing in ceremony of Indian Prime Minister Narendra Modi as an honored guest of the Bharatiya Janata Party," says Sangay's Facebook page.
Tenzin Gompo wrote: "I saw Sikyong on TV. He just sat behind Mr L.K. Advani and beside to Mr Akhilash Yadav, chief minister of Uttar Pradesh. Sikyong was also a honoured guest."
"Wow it's great honour to our Sikyong and Tibetan government in exile Jai Tibet, Jai Hind," said Pema Cescten.
Tashi Stobgyas wrote: "Thank you India for your respect towards Tibet."
"Yes, for a very short fleeting moment I saw someone like our Sikyong among the guests, but the clip was not repeated. So I was not sure. Now I am convinced," said Dawa Dhargye.
Sangay took over the reins of the exile government in August 2011 from 74-year-old monk scholar Samdhong Rinpoche, who held the post for the past 10 years.
With the Dalai Lama stepping down from diplomacy and active politics, the elected leader of Tibetan people has acquired added stature.
The Dalai Lama, a Nobel laureate, has lived in India since 1959 when he fled his homeland after a failed uprising against the Communist rule. The Tibetan government-in-exile is based here but is not recognised by any country, including India.