With the men's curling team in the final against Canada on Friday, Britain are now guaranteed matching their best ever Winter Games medal total of four.
Muirhead, the leader of an all-Scottish team, punched the air in delight after she sealed victory with the last stone of the 10th end, keeping her cool to take the point that left the Swiss empty-handed.
"When you look at our success over the last couple of years, the one medal we were missing was the Olympic medal and we have got that now," said Muirhead.
"I am so happy, we are all pushing each other and it makes it more fun to win something," she added.
Britain's women had won the curling gold in 2002 when Swiss skip Mirjam Ott was one of the defeated finalists.
Britain took the lead on Thursday for the first time in the eighth of the 10 ends when an official's measuring gauge was needed to grant them a second point.
Ott tied it up at 5-5 with her last stone in the ninth end to set up a tense finish.
For Ott it was another disappointment after silver medals in both 2002 and 2006 and a fourth place in 2010.
"It's not the right time to say, there are too many emotions right now," Ott, 42, said when asked if she would continue in the sport.
(Writing by Keith Weir; Editing by Ed Osmond/Peter Rutherford)
