Florence Schelling made 28 saves for the Swiss, who earned their first medal in the sport that was added to the Winter Games in 1998.
Valentina Wallner stopped 22 shots for Sweden.
Canada will play the United States for the gold medal later today at the Bolshoy Ice Dome.
Sweden scored once in each of the first two periods before the Swiss tied it with 13:47 left in regulation on goals by Sara Benz and Pheobe Stanz.
Switzerland made it 4-2 on an empty-netter with 67 seconds left before Sweden made it a one-goal game.