The Indians had a one-set lead but Venus Williams stepped up brilliantly with her partner Rajeev Ram to win the second set and clinch the issue 2-6 6-2 10-3 in the super tiebreaker.
A four-time Olympic gold medalist Venus will now aim for a fifth and first in mixed doubles.
"We have to try and recover as quick as possible, mentally and physically. Probably more mentally. We have to go back, try and get some sleep, put some food in our body and recover for tomorrow," Sania said after the semifinal defeat.
The world number one doubles player Sania said the challenge would be to overcome mental fatigue as she seeks her first Olympic medal from three appearances.
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The US pair broke Sania's service twice in the fourth and eighth game of the second set.
"Tennis is a game of inches and cms, anything can happen but we are coming back. She did step up well, that's why she's a champion. She's able to do that under pressure. She started returning a lot better as opposed to the first set.
Giving credit to Venus, Bopanna said, "Once they got that break (in the fourth game of second set), Venus started serving much better. The good thing is we still have a chance to come back and get a shot at the bronze medal. We have to recover well and come back by being positive. Our opponents will also be coming from a loss.