The 28-year-old Scot, nicknamed The Highlander, came home third in the men's slalom at Salt Lake City.
Baxter finished eighth in the first run, climbing to third in the second to finish 1.26 seconds behind French winner Jean-Pierre Vidal.
Vidal's compatriot Sebastien Amiez was second.
Baxter's shock success was Britain's third medal of the Games, following the women curling team's gold and skeleton slider Alex Coomber's bronze.
The three-medal haul is Great Britain's best result at the Winter Olympics since 1936.
And Baxter's performance was the best British alpine result since Georgina Hathorn came fourth by 0.04 seconds in the women's slalom in 1968.
The previous best by a man was Martin Bell's eighth place in the 1988 downhill.
Baxter, from Aviemore, had been struggling with his form and ski equipment in the build-up to Salt Lake City.
After claiming bronze, he told BBC Sport Online: "It's unbelievable to think I've won the first alpine medal for Britain at an Olympics, and I am sure there will be more to come.
"All the hard work over the season has paid off.
"If I had put two really good runs together, I would have won gold."
Baxter made headlines earlier in the Games when he was forced to change his hairstyle, after dying his hair blue and white in the shape of Scotland's national flag.
He ran conservatively on his first run in the slalom, but claimed an all important top-15 spot for the second.
On his second run he initially appeared to have lost a place when he was slower than Amiez.
However, those who followed struggled on the testing "Know You Don't" course.
"That slalom was unbelievable"
Jubilant Baxter on his bronze medal
Norwegian Kjetil Andre Aamodt was slower and home favourite Bode Miller sensationally crashed out to assure Baxter of his medal.
Vidal, the leader after the first run, came down last to claim gold and move Baxter down into third.
Alain's younger half-brother Noel finished 21st, with team-mate Gareth Trayner a further two places behind.
Speaking later about his medal, Alain said: "It's pretty cool. I struggled all year with my equipment and I was losing confidence.
"But I knew after last season's results that it was possible.
"This does exceed my expectations a little bit.
"It's what I have been training for all my life. I've put a lot of my time into skiing and it's paid off."