Delighted friends and fans cheered on The Highlander as he made history in a dramatic slalom event.
Baxter, 28, improved from eighth position after his first run to win the bronze medal behind Frenchmen Jean-Pierre Vidal and Sebastien Amiez.
Watching, and cheering on, from the sidelines were a bunch of 'Tartan Army' supporters, including several fellow members of the Great Britain squad.
It was very much a family affair as Baxter celebrated his unique achievement.
His cousin Lesley McKenna, who took part in the snowboarding at the Olympics, told BBC Sport Online: "This is an unbelievable day.
"I was much more nervous for him than I was competing in my own event."
McKenna had returned to Salt Lake City to watch her cousin after taking a break following her competition.
"I'm so pleased I came back to the Olympics. I wouldn't have missed this for the world," she said.
"Our grandparents May and Chick Baxter will be going wild back in Scotland."
Alain's younger brother Noel, who finished 21st in the slalom, also paid tribute.
"I watched his race before I competed, and thought 'great, we are probably going to get a top five, which would be a fantastic result'.
"When I knew he had got a bronze medal, it was amazing.
"Then I had to think about doing my second run. It was hard to concentrate because I was so excited."
Fellow Brit Gareth Trayner, who finished 23rd in the event, said Baxter's success would be a huge boost for the sport.
"Alain's been struggling all season, and it couldn't have happened to a nicer guy.
"We will certainly be partying tonight."