Switzerland appeared to have the upper hand in the early stages, but the decisive moment of the contest came in the seventh end, when a brilliant takeout from Bruce Mouat cleared away four stones and forced the Swiss to relinquish the hammer for just a point.
From then on, Britain did well to restrict their opponents and though Switzerland had the last stone of the game, Benoit Schwarz-van Berkel was not able to get the connection he needed.
For Switzerland, the first men's team to go unbeaten in their round-robin campaign at the Olympics since Kevin Martin's Canada in 2010, it was a crushing defeat after what had been an excellent week.
Britain and Switzerland also met in the final of the world championships last year, when Team Mouat took a 5-4 victory.
"We weren't having our best in the first five ends and then really flipped it around in the last five. So just really proud of us for sticking together, supporting each other and believing in each other," Mouat said.
Canada beat Norway at second time of asking
Two sheets over, Canada's Brad Jacobs, who won gold at the 2014 Games in Sochi, navigated his team through a defensive battle to prevail 5-4 over Norway.
The game went to an extra end after Jacobs' attempt for a double takeout only got rid of one stone, but when Norway handed over the hammer with their two-pointer, it was easy enough for Canada to inch ahead and get the win.
The two teams had met just hours before in the final round-robin session of the men's competition, in which Magnus Ramsfjell's Norway were 8-6 victors, though it was not against a full-strength Canada line-up with Ben Hebert rested.
"They were tough all day. Magnus made a ton of great shots, starting in that first game. Those guys were virtually lights out," Jacobs said.
"They gave us everything we could handle. I was just grateful that we had hammer in that game, and we got off to a good start. We were able to manage the scoreboard, because they were not going away easy.
"They played awesome. Just happy to be able to get through them in that semi-final."
Britain will meet Canada in the gold medal game on Saturday, while Switzerland and Norway will face off for bronze on Friday.