Alex, an RAF officer, was favourite for the top prize of gold after her first run.
But she was narrowly beaten into bronze by US girls Tristan Gale and Lea Ann Parsley.
She had to race through heavy snow at the Utah Olympic Park track, but managed a good time in her first run of 52.48 seconds.
Her second run saw her move to first place, but in the end the US pair had better times in their second runs after her.