The mixed doubles third seeds saw their hopes of winning gold ruthlessly crushed by England's Chris and Gabby Adcock early on Saturday at the Emirates Arena.
But they rallied for the evening's third-place play-off, producing a dominant performance of their own to beat Malaysia's Peng Soon Chan and Lai Pei Jing 21-17 21-11 and spark jubilant celebrations.
For Glaswegian Bankier it represented justification for a controversial decision to quit the Badminton England-run Great Britain set-up after London 2012 and return to Scotland ahead of the Games.
The 26-year-old cried after the final point was won and embraced her proud father Ian, chairman of Celtic FC, as she left court. Her success was all the more poignant coming barely two minutes' walk from Celtic Park.
Bankier said: "The season building up to this week has been very stressful at times. We have had amazing highs and terrible lows, all for this tournament.
"Losing this morning was incredibly disappointing, so to pick ourselves up from that and come out and perform this evening was a challenge.
"But we are so happy and it is very special to do it in front of a home crowd."
Bankier and Blair's clash with the Adcocks, the top seeds, had been billed as one of the matches of the tournament but it failed to live up to expectations as the top seeds won 21-14 21-14.
The Adcocks will now play team-mates Chris Langridge and Heather Olver, 2-1 victors over the Malaysian pair in their semi, for gold.
Blair said: "We had to pretend the semi didn't happen. It was difficult to come back and perform, but the calmest I'd been was being back on court.
"The bronze medal match is the last-chance saloon, you don't want to go away with nothing.
"To do it here is by far the most satisfying thing I have done in my career."