The two sides traded blows right through until the sixth end, with the Swedes 5-4 to the good heading into the closing stages, still with work to do to round off the contest.
But they did so in some style in the very next end, as skip Niklas Edin drew his last stone right into the house to score five points and end the match there and then at 10-4.
"We started well but couldn't quite figure out the ice since it's changed a bit from game to game," said third Oskar Eriksson.
"We were also a bit tired out there both mentally and physically, but we said during the fifth end break that we just need to stay focused and keep grinding to stay in the lead.
"Now it feels really great. By winning this key match it gives us the chance still for the 1v2 page play-off game"
There was further drama in the afternoon session as Norway stole victory over Germany, in a game that went to the extra end.
With the final shot of the game Germany's skip Alexander Baumann attempted a near impossible run back in, missing the opportunity and handing a 8-7 victory to their counterparts.
That was Germany's second defeat of the day, also going down 8-5 to USA who made it two wins from two with a confident victory over China.
USA skip John Shuster tapped up one of his own stones to score two points and give his team a 5-1 lead after the fourth end, with a Chinese mistake seeing them steal two more immediately after, eventually prevailing 8-2.
Scotland beat Italy 6-2 in the evening's final game, with the Italians also losing to Switzerland earlier in the day, while the Netherlands picked up their first win against fellow strugglers Russia, 6-4.
Read the original article on Eurosport: Sweden reach World Championship play-offs after victory over Switzerland