The world No.1 saw her hopes of winning a gold medal for Australia at Tokyo 2020 brought to an end on Friday.
Barty and partner John Peers won the first set against Russian Olympic Committee duo Andrey Rublev and Anastasia Pavlyuchenkova.
But Rublev and Pavlyuchenkova turned it around to win an epic clash 5-7 6-4 13-11 and reach the final.
With Barty already eliminated in the singles and women's doubles competitions, the best she can now hope for is a bronze medal alongside Peers.
They will meet Serbia's Novak Djokovic and Nina Stojanovic for the last spot on the podium.
"It is a tough one to swallow," Barty said. "I think those tie-breaks at times, they go either way and we just didn't quite have the run of the grain.
"We put ourselves in a position to win the match, we just weren't able to close it out. It was a good level, no doubt disappointing."
Another chance at a medal means Wimbledon champion Barty will try to get over her loss swiftly.
She added: "But we get to fight for a medal – this one hurts but we still have an opportunity to play another match and try to win a medal for Australia.
"We pick ourselves up pretty quickly, move on and know that we will give it our all and keep fighting right until the end.
"It's been quite a while since we've had an Australian medallist here [in tennis] at the Olympics so we're looking forward to that challenge."
In the men's doubles, Nikola Mektic and Mate Pavic are Olympic champions after they beat fellow Croatia representatives Marin Cilic and Ivan Dodig in the final.