The Finns built a two-goal lead twice and held off the Russians in the third period after they had narrowed the margin to one.
Petra Nieminen scored in the first period for the Finns, netting her third goal of the Games, and Susanna Tapani doubled the lead early in the second period.
Russia's Olga Sosina narrowed the margin to 2-1, but Linda Valimaki restored Finland's two-goal cushion later in the period.
Lyudmila Belyakova tallied on a Russian powerplay to make it 3-2 in the third and end the scoring.
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Finland have now won three Olympic medals in women's ice hockey, all of them bronze. They last earned a medal in Vancouver in 2010 and their first came in Nagano in 1998 when women's hockey made its Olympic debut.
The gold medal game is set for Thursday, a showdown between Canada and the United States who are the only two countries to have won Olympic gold in the sport.
(Reporting by Dan Burns, editing by Ed Osmond)
