Having a shot at any colour of medal looked beyond the heavily-touted British men as recently as Wednesday morning.
But their most accomplished performance to date – against the United States – served notice of their threat and they came into the matches that matter most, stress-tested in a way the Swiss had not been.
Indeed, the closest Yannick Schwaller’s men had come to defeat here was in being taken to an extra end by the GB quartet on Sunday.
But Switzerland had played on this sheet earlier in the day and initially adapted better to the conditions, taking two in the second to cleave open an advantage.
In other matches this week, a slow start has cost the British rink but they responded much better this time, Mouat levelling the scores with a composed draw in the third.
These are arguably the two best teams in the world right now, but mistakes were playing a part in a contest that lacked the high-grade quality of their earlier meeting.
A Mouat misjudgement allowed the Swiss to take another two in the fourth and open a 4-3 lead at the break, but the errors were coming from both sides.
And in the sixth, we had one that changed the whole tenor of the contest.
Switzerland had a shot at three with their final throw. But the attempt glanced one of their own stones on the way through and skittered right through the house to give Britain a steal and fresh hope at 4-4.
The Swiss nudged back ahead in the seventh, but only by one after a fabulous Mouat shot averted a curling catastrophe, and the GB skip then added two of his own to the board to give his rink the lead for the first time with two ends left.
What did the Swiss have in response? They opted to blank the ninth, meaning two in the final end would put them in the final and given them revenge for their World Championship final defeat last April.
It came down to the final stone but Benoit Schwarz-van Berkel was a fraction off and that fraction has secured at least a silver medal for Team GB.
“That’s probably the hardest-fought win we’ve ever had and glad we brought our best when it really mattered,” second Bobby Lammie told BBC Sport.