Sania and Bopanna, who are being touted as India's medal hope at the Olympics, broke little sweat in their opening round match to post a 7-5 6-4 win in 73 minutes.
The fourth seeded Indian pair took a little time to settle down but once they got the rhythm, there was hardly any looking back.
"Personally for me it would be amazing to win a medal at the Olympics, which I don't have. It would mean the world us.
We would do whatever we can and put our best forward," the world number one women's doubles player Sania told reporters after the match.
On a chilly and windy evening at a small court of the Tennis Centre in Barra, a lot of Indian supporters waited patiently since 3.30pm local time for the match to begin. It was delayed by men's doubles semifinals featuring Rafael Nadal and Marc Lopez, that took more than two hours to finish.
Waiting among the 'spectators' were Leander Paes, Sports Minister Vijay Goel and SAI director Injeti Srinivas.
In the first set, both teams did not drop a point till the ninth game before Peers' serve was broken by the Indians and they went 5-4 up. But it did not help Sania and Bopanna's cause as they conceded the advantage in the next game itself and were broken back with the scoreline levelled at 5-5.
But before it went out of their hand, the Indians capitalised on an unforced error from their rival to take four straight points and break Stosur's service and wrap up the first set in 36 minutes.
In the second set, the Aussie pair started to wilt under pressure and a double fault from Peers gifted the Indians an early break and they took a 3-2 lead.
Even as 'come on India' chants grew louder, Sania and Bopanna held on to their serve to surge ahead 4-2. Bopanna sealed the issue in the 10th game with three big aces, to which the Aussie had no answer.