British skip Eve Muirhead secured victory with a draw to the button on the last stone.
Muirhead's world champion rink recovered from a 6-4 loss to Canada in Wednesday's semifinals to win Britain's first curling medal since 2002, when the women's team captured the gold.
Britain already had won a gold and a bronze in Sochi and is assured of either a gold or silver in men's curling with David Murdoch's team in the final. Britain last won four Winter Games medals in 1924.
Jennifer Jones and her 10-0 Manitoba rink play Sweden for the gold medal starting at 8:30 a.m. ET.