They joined Edinburgh’s Bruce Mouat and the men's team in guaranteeing a medal after overcoming a nightmare start at the National Aquatic Centre.
Muirhead's attempted draw with her last stone of the first end finished almost two metres short and gave Swedish skip Anna Hasselborg the simplest chance to roll in for a four.
Immediately it looked like the Britons might be facing a repeat of their heavy semi-final defeat to the same team in Pyeongchang four years ago, when they failed to recover from a poor start and lost 10-5.
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But Muirhead hit back strongly by taking three points in the next to give her team hope, and two single ends in succession hauled her back level at the halfway stage, albeit with Sweden retaining the 'hammer' advantage.
Hasselborg pressed back in front with a two-point end in the sixth, but errors were becoming more prevalent in the Swedes' play and a missed double take-out attempt in the next gave Muirhead the chance to roll in for two to level at 7-7.
A packed eighth end resulted in Hasselborg having to settle for taking a one, before Muirhead produced a magical shot to take four in the ninth.
Hasselborg responded in kind to brilliantly take three and force an extra end. Britain held the 'hammer', or final stone, but Muirhead did not need it as Hasselborg's attempt with her final stone drifted wide and gifted the Britons the point they needed to seal a stunning win.
It means the 31-year-old Muirhead will experience her first Olympic final at the fourth time of asking, along with team-mates Vicky Wright, Jen Dodds from Edinburgh and Hailey Duff.
Japan await in the gold medal match after beating Switzerland 8-6.