Call it the Cortina Comeback. At nine o’clock on Thursday morning, Great Britain’s men’s curling team didn’t even know whether they would be in the Olympic semi-finals. By nine o’clock on Thursday night, they had won themselves a place in Saturday’s final against Canada by beating the undefeated Swiss team 8-5.

The British, who scraped through the round robin only because the Italian team lost their last group stage game to the Swiss on Thursday morning, had promised that they would be an entirely different proposition if they got to the knockout rounds, and they were as good as their word.

“An emotional rollercoaster is probably the way to describe this whole week, never mind just that game,” said Hammy McMillan. “I guess I’m still in a little bit of shock, because we had to really fight extremely hard in that game. We started off pretty slow and the Swiss guys really took advantage of that.”

It was a hell of a game, played in front of a rowdy crowd, the loudest of them a happy band of mad Scottish curling fans who had come along with a repertoire of songs written especially for the four men on the team. There was a lone bagpiper in there somewhere too, and even someone playing a kazoo.

“They were amazing. If they can bring that exact same energy on Saturday, it’ll spur us on,” McMillan said. “Any time you hear the bagpipes, you get goosebumps.” Well, yes, that’s one way of describing it.

The game started to go their way in the sixth end, which the British managed to steal when Switzerland’s vice skip missed a tricky hammer shot. But it really turned on an improbable shot by Bruce Mouat in the seventh end. The curlers call it a “run back triple takeout”, but the rest of us might understand it better as a feat of geometry in which Mouat managed to hit one of the opposition’s stones into a second in such a way that the second scudded into a third. The Swiss, who had been expecting to reap at least three points and move into a 7-4 lead, suddenly found they needed to make a draw just to score one.

Bruce Mouat on the tense final end against Switzerland which Britain won to seal the passage into Saturday’s final. Photograph: Issei Kato/Reuters

Mouat judged it one of the best shots he has ever made in competition. You could tell he liked it because he even allowed himself to celebrate with a little fist pump. “In my opinion Bruce is one of the best shotmakers in the world, if not the best shotmaker in the world,” McMillan said. “You know, for the last 18 months he’s been incredible, and as long as we do our job, we’ve got 100% confidence in Bruce he’s going to finish it off.” Britain were able to score two in the next end, which meant they led 6-5 going into the 10th.

Mouat and the Swiss skip Yannick Schwaller have a friendly rivalry that runs right back to their matches in the world junior championships a decade ago. Schwaller won the first games they played but Mouat has had much the better of it in the years since, and led 22-10 in their head to head coming into this semi-final, including their last four games straight. Last April the British beat the Swiss 5-4 in the world championship final.

But it was around that time that the Swiss brought in the legendary Canadian curler Glenn Howard, a four-time world champion, as their national coach. Howard has built the Swiss team into the machine that was unbeaten in the round-robin stage of these Olympics, and during the time-out he came on to the side of the ice to try to steer his team through the final shots they needed to make to take that 10th end. But Mouat’s team had their measure. The Swiss missed with their final stone, and the British broke out into such wild celebrations that Bobby Lammie nearly clattered the referee with his broom when he threw it into the air.

Now Mouat, who has played 16 days straight at these Olympics, has one game left. “It’s going to be a tough game,” says McMillan. “We play Canada a lot. We beat them in the semi-final of the world championships last year, and we know it’s going to be a battle. On paper this week they’ve been the second best team behind Switzerland. So we know we’ve got to bring the way we played in the last four or five ends there today for all 10 ends. We haven’t probably quite had a full A+ 10-end game yet. So it’s time to bring it on Saturday.”

The British women weren’t so lucky. They beat Italy 7-4, their third-straight victory, to give them a 5-4 record for the tournament, which was good enough to get the men through, but they needed Switzerland to beat the USA too if they were going to make it. The Swiss came a couple of inches away from doing it, which seemed pretty unlikely when the US team were 6-3 up going into the 10th end. The Swiss skip pulled off an improbable shot to equalise the game with the final stone, only for the US to win in the extra end.