At 33, Jones was the oldest entrant in the final, but she showed her experience with her best run of the week on her last attempt.
Her score of 87.25 briefly put her top of the standings and although she was overtaken by Finland's Enni Rukajarvi (92.50), who took silver, and gold medallist Jamie Anderson (95.25), her third-place finish is a wonderful achievement.
Scottish skier Alain Baxter won the first GB medal on snow in the 2002 event at Salt Lake City, but was later stripped of the bronze after failing a drugs test. Baxter said later the drug came from a US version of a cold remedy inhaler.
Jones' bronze on the second day of the competition is Britain's earliest medal in the Winter Olympics.
Less than a day on from Jamie Nicholls and Billy Morgan earning top-10 finishes in Saturday's men's final, Jones will now be aiming to continue the feelgood factor over snowboarding in Britain.
In Thursday's qualifiers, she became the first woman to compete in the Olympic snowboard slopestyle event, which is making its debut in Sochi, and finished fifth, requiring her to compete in the semi-final.
But today, she was the last out of the gate and knew what was required of her before setting off.
She benefited from several people failing to land their jumps, and although she initially struggled on the rail part of the Rosa Khutor Extreme Park course, she completed three excellent jumps to move into third after the first run.
Qualification was already secure by the time of her second attempt, although she did slightly improve to take her place alongside Sarka Pancochova (90.50) of the Czech Republic, Swiss Sina Candrian (84.25) and Norway's Silje Norendal in the final.
"I'm just chuffed," she said.
"It was a bit nerve-wracking having to drop last and then when I realised I'd done it, it was a good feeling."
The Jenny Jones factfile
1980: Born July 3 in Bristol.
1999: Explodes onto the scene by winning the first of five British Snowboard Championships.
2006: Finishes the year second in the World Snowboard Tour Rankings.
2009: January - Cements her position as one of the world's leading snowboarders by winning a slopestyle gold medal at the prestigious Winter X Games in Aspen, Colorado.
February - Earns silver in the slopestyle at the inaugural Winter Dew Tour after second-place finishes in the meetings at Breckenridge Ski Resort in Colorado and Mount Snow Resort in West Dover, Vermont.
2010: January - Wins her second slopestyle gold medal in a row at the Winter X Games 14 in Aspen.
February - Runner-up once again in the Winter Dew Tour standings after a third-place finish at Breckenridge was followed up by coming second at Snowbasin Resort in Huntsville, Utah and Mount Snow Resort.
March - Follows up her success in the X Games with yet another gold medal at the Winter X Games Europe in Tignes, France.
2011: Fails to claim a hat-trick of Winter X Games gold medals, settling for silver after being pipped by Finland's Enni Rukajarvi.
2013: August - Secures her first World Cup podium with a silver medal in a meeting in New Zealand.
December - Suffers a concussion in a training crash in Austria.
2014: January - Named in Team GB's squad for the Winter Olympics in Sochi to compete in the Games' first ever slopestyle event.
Sunday, February 9 - marks her Olympic debut at the age of 33 by winning bronze medal for Great Britain in snowboard slopestyle at Sochi 2014 Winter Olympics. Becomes Britain's first medal winner on snow.