RAF officer Coomber, who was the hot favourite for gold, finished behind American duo Tristan Gale and Lea Ann Parsley.
Coomber said: "I'm ecstatic, that was absolutely brilliant... I can't be disappointed with an Olympic medal.
"This is indescribable - it's the most nerve-wracking experience of my life"
Alex Coomber after finishing third
"The worst moment is when you come over the line and you don't immediately know what your position is.
"I had a bump on the 11th curve, I've been having trouble with that all week.
"The snow didn't help, but I was not too worried, I knew I could still go fast," the 28-year-old added.
Heavy snow, which can cause problems with momentum, greeted Coomber's arrival at the Utah Olympic Park track.
She made light of the difficult conditions by posting a creditable time of 52.48 seconds in the first run.
That was enough to push her into second place, but her time was later eclipsed by Lea Ann Parsley, who finished just 0.01 of a second behind her American compatriot Gale.
The weather and track conditions suited the heavier sliders better and Coomber could not improve on her third place in the second run.
Sports Minister Richard Caborn was delighted for Coomber and said: "This is great news.
"Congratulations to Alex Coomber on winning Britain's first medal at Salt Lake City in one of the newest Olympic sports."
Women's skeleton final - combined finishing times:
1) T Gale (USA) - 1:45.11
2) LA Parsley (USA) - 1:45.21
3) A Coomber (GB) - 1:45.37
4) D Sartor (Ger) - 1:45.53
5) M Pedersen (Swi) - 1:45.55
6) L Alcock (Can) - 1:45.69
7) S Hanzlik (Ger) - 1:45.95
=) E Mironova (Rus) - 1:45.95
9) D Locati (It) - 1:46.65
10) M Kelly (Can) - 1:47.32
11) L Couch (NZ) - 1:47.80
12) E Nakayama (Jpn) - 1:48.72
13) C Ninos (Gre) - 1:49.28