Olga Sosina answered for the Russians but Linda Valimaki responded for Finland later in the second period. Lyudmila Belyakova's power-play goal with 13:57 remaining pulled the Russians within the final margin but they could not find an equaliser.
The Finns matched their best Olympic women's hockey finish, having also taken bronze in 1998 and 2010. Russia have never taken a medal in the event. Four-time defending champion Canada and the United States meet Thursday.