Britain, led by skip David Murdoch, began their match against reigning world champions Sweden, knowing victory on the ice would hand them a guaranteed medal.
And with so much at stake, it proved to be a tense game with the lead switching hands before a last-end Swedish error handed Murdoch a chance to send down his final stone and seal a 6-5 triumph.
He said: "Incredible, I cannot describe the feeling. We have trained so hard for this and made a lot of sacrifices and it has paid off.
"You always believe and work hard, beating yourself up in training to be here. I am so proud of the guys and we have worked tirelessly for the last few years."
They will come up against Canada in Friday's final after the reigning Olympic champions defeated China 10-6 in the other semi-final.
Leading 7-6 after eight ends, the Canadians secured three in the ninth leading to China to concede defeat and instead a battle for bronze against Sweden.