The announcement came after India's last medal hope Yogeshwar Dutt lost 0-3 to Ganzorig Mandakhnaran in the qualifying of the 65kg freestyle wrestling.
Sakshi, who had become the first woman wrestler in India to win an Olympic medal with her bronze in the 58kg freestyle, became the obvious choice as India's only other medallist PV Sindhu, who bagged a silver in women's singles badminton, left for Hyderabad on Saturday.
"Sakshi and Sindhu have been the medallists for us and both displayed immense courage and determination during their competitions," said chef de mission Rakesh Gupta.
"Sakshi started the account for us and later Sindhu became the first Indian woman to win a silver medal. Sindhu, in fact came close to making it a gold. Both Sakshi and Sindhu are young and have many more Olympic Games ahead of them, we can look forward to even greater performances from them.
"Sakshi opened the medal account for India in Rio and Sindhu added to it. These young ladies have been an inspiration for all women in India and I am sure many more will follow them," Gupta said.
N Ramachandran, President of the Indian Olympic Association, came to the Olympic Village and wished the two medallists and their coaches.
"I met Sindhu and Gopi and also Sakshi and her coach, Kuldeep. They have been great performers. I also congratulate others who came close to medals, like Abhinav Bindra, Dipa Karmakar and Sania and Rohan Bopanna.
"They were good enough for medals, but it was not their day. I wish them and all the others for their efforts and we hope that we can do even better in the future."
The Games will close a circle with Olympic flame being extinguished and apart from the melancholy the organisers have promised to show the best of Brazil.
According to Rio 2016 communications director Mario Andrara, 41,233 tickets were sold for the Closing Ceremony at the Maracana.
However, it was still unclear whether Brazilian legend Pele who had pulled out of the Opening Ceremony would attend the function.
"If he wants to be there, he will be our guest, but we haven’t got in touch with him or he hasn’t got in touch with us since the Opening but obviously if we get a chance to have him it will be our pleasure," Andara said.