Finnish captain Jenni Hiirikoski scored the winner when she stretched out on the ice to tap a long pass past Swiss goalie Florence Schelling, who made 30 saves in the Group A loss.
Switzerland, who were outscored 14-0 in their previous two preliminary round games, erased an early 2-0 deficit and then forced overtime when a shot from behind the net bounced in off a Finnish defenseman with under four minutes left in regulation.
The four Group A teams are given a free pass to the knockout phase due to a new format that ensures the top nations advance. The win means Finland, beaten 3-0 by favourites Canada on Monday, next play the lower of the two Group B seeds. Switzerland will face the Group B winners.