Conway had come agonisingly close to beating Colombian judo legend Yuri Alvear to make it into the gold medal match, but despite dominating the bout just couldn't score - and ended up losing out to a 'golden score' from the three-times world champion.
The British judoka shrugged off that disappointment, however, beating Bernadette Graf of Austria by a single point to win the bronze medal match.
"I knew if I was going to come away with a medal I would have to put in my best performances.
"I'm not from Brazil but the crowd have been awesome. I just want to thank everyone....
2016 Rio Olympics - Judo - Final - Women -70 kg Bronze Medal Contests - Carioca Arena 2 - Rio de Janeiro, Brazil - 10/08/2016. Bernadette Graf (AUT) of Austria and Sally Conway (GBR) of Britain compete.
Image credit: Reuters
"This has been my dream since London. I can't put it into words."
Alvear went on to lose the gold medal match, with Japan's Haruka Tachimoto winning by Ippon. Laura Vargas Koch also won a bronze medal, having come through the opposite side of the draw to Conway.
In the men's judo, Mashu Baker made it a second gold of the night for Japan in the dojo as he claimed the men's -90kg gold.