David Murdoch's all-Scottish rink, one of the main British hopes for a medal in Russia, had been beaten by the Norwegians in their round-robin fixture and played catch-up for most of the game, until his brilliant final stone in the 10th end secured two game-winning points.
GB will now play Sweden on Wednesday night, with the winners taking on either China or Canada in the final on Friday.
Murdoch's final throw, a tricky double take-out, will go down in curling history should GB go on to win a medal.
It came after a tension-building time-out where Swedish coach Soren Gran came on to the ice to help the British rink to decide whether to go for the shot or hand advantage back to the Norwegians at the extra end.
Murdoch described the win as "incredible" and admitted the delivery, "will certainly go up there as one of my best"
"I stood back and saw there was a shot for two and once we seen that, we were always going to go for it," he said.
"Sometimes you just have to go for it. You have to trust everything you have done previously, all your practise shots, everything you have done all week.
"We had the courage to go for it and were confident we were going to take it.
"I have played those guys a hundred times and the chances of stealing an extra end is very slim.
"So, as hard as the shot was, we had to go for it.
"There was no margin for error. If you are an inch wide, you don't get it, so we were perfect on that one. It wasa fantastic team effort.
"It's so great to be in the semi-final.
"Today has set us up for Sweden. Those guys obviously have had a great tournament so it's important we go in there with a win."
Vice-skip Greg Drummond confirmed the difficulty of the task that lay ahead of Murdoch as he surveyed the crucial final throw.
He said: "You're going to make that shot once in 50, maybe two in 50. It's a difficult shot, especially to win the match.
"In the middle of the game, if you have a go at it and miss so nobody thinks anything of it as it is a tough shot.,
"But to win the match with it, it's probably one of the best he has played all season.
"We called time out and weighed up our chances to steal the extra end.
"Against Norway I would rate the chances at 35 per cent on a good day so it was probably worth going for a win."
Gran was confident that Murdoch was in the right frame of mind to come up with a "spectacular shot."
"I said that I think we should try it, to go for the double," he said.
"David looked at me and smiled and that was the most important thing for me, to see that smile and see that fighting spirit.That shot meant a lot."
Norway skip Thomas Ulsrud could barely believe his team had been knocked out.
"First of all, all credit to GB for sticking in there," he said. "I think we played a good game, were in control of the game all the way.
"And after the last rock came to rest, I looked at the scoreboard and said, 'They actually won that one.'
" It didn't feel like that, (I was thinking) this is not happening. Tell me it's a dream. I thought we played solidly the whole week.
" If you ask me, I thought we played better than fifth."