They claimed a tense 6-5 semi-final win over odds-on favourites Canada.
In Thursday's final, Britain will meet Switzerland, who defeated the US 9-4 in the other semi-final.
The defeat of Canada is already being hailed as the greatest win in British curling history.
And it capped a medal-winning day for Team GB, who had drawn a blank in the Games until then, after women's skeleton slider Alex Coomber won bronze in her event.
"We will have to play well if we want to win"
Martin on final opponents Switzerland
It guarantees Britain's best medal haul since 1948, when they won two bronzes.
Rhona Martin's team thought they had been eliminated when losing to Germany on Monday.
But they beat Sweden, and then gained revenge over Germany, in two play-off matches to reach the last four.
And now they have completed a remarkable revival by defeating the reigning champions Canada.
Martin's team held a 5-3 lead heading into the eighth end of the semi-final.
And they kept their nerve when the Canadians came back to tie at 5-5 going into the 10th and final end, with Martin's last stone proving decisive.
Reigning world champions Canada had lost just one of their eight preliminary round games and were regarded as being virtually unbeatable.
Afterwards, team coach Russell Keiller talked about the bravery of the British team.
"To come back the way they did yesterday was great and then come out and beat the favourites in the semis was tremendous," he said.
"It's ironic that we're playing Switzerland now"
National coach Mike Hay
"We were confident in the tenth end because Rhona could see the four-foot ring and she played the right shot."
National coach Mike Hay added that it was strange that their final opponents had done them a favour, earlier in the competition.
"It's ironic that we're playing Switzerland now - it was their win against Germany that kept us in the tournament," he added.
Curling skip Martin kept calm despite the fact that her team are now just one step away from making history.
"When we stole the two in the fourth end that was really good, because it turned the hammer, and we just kept piling the pressure on.
"The ninth end was a bit scary but as long as we kept the hammer and hadn't lost the two, that was fine. We made Canadian skip Kelley Law play the shots."
"It'll be tough, they're a really good side and we'll have to play well," she added, looking forward to the final.