The teams traded deuces through nine ends, but Sweden crucially had the last stone in the 10th.
It fell to the final shot, when Edin needed just to draw his stone into the four-foot ring. The team over-swept and the stone went further than expected, coming to rest just in time.
"When we shook hands, I had to ask them who won," Edin said. "It was kind of a weird moment not knowing."
Sweden also beat the Canadians in the final of the 2013 world championships, held in Victoria, Canada.
Elsewhere, Norway continues to show there is more to its curling than its funky dress sense.
Against Russia (0-3), the pattern on the Norwegians' pants was red-and-blue splattered paint, meaning they still haven't worn the soccer socks/flat cap/knee-length knickers combo that wowed observers during practice sessions over the weekend.
Norway was always in control against the host nation and the final score slightly flattered the Russians, who picked up three points at the last end.
Germany, the United States and Russia are the three teams without a win heading into Day 3.