Selanne, a 43-year-old, six-time Olympian, and Jussi Jokinen scored 11 seconds apart early in pivotal second period after a scoreless first. The Finns had three goals in the third against a team that looked like it would rather be at home.
Finland has won four medals in the last five Olympics, more than any other nation in the NHL era.
Before the third-place match became a rout in the final period, it was a game of missed opportunities for the Americans.
Patrick Kane couldn't convert on a penalty shot in each of the first two periods for the United States. He missed the net to the right on his first one-on-one duel and hit the right crossbar on his second.
The Americans wasted a chance to earn medals in consecutive Olympic hockey tournaments for the first time since winning gold in 1960 and silver in 1956.
Finland, meanwhile, took advantage of two power plays in the third period with goals to put away the United States in game it didn't look interested in competing in after falling behind 2-0.
If the league and players' union do not let the world's best players go to South Korea in four years, Kane & Co. may never get a shot to help the Americans win gold that has been elusive since the "Miracle on Ice," in 1980 when it beat Finland after their historic upset of the mighty Soviet Union in the semifinals.
Jonathan Quick, starting ahead of silver-medal winning goaltender Ryan Miller, stopped all eight shots that got to him in the first before giving up five goals on just 21 shots over the last two periods.
Selanne, who has said he will retire after playing for the Anaheim Ducks this season, skated off the ice in Sochi for the final time with 20 seconds to play and hugged two teammates on the bench before leaping back over the boards at the final buzzer.
With bronze medals around their necks, Selanne, a first-time Olympic captain, led his team on a victory lap around the ice as the players threw bouquets of flowers to fans before posing for a picture at center ice.
The United States seemed set up to end its Olympic hockey championship drought, or at least come home with a medal, with players seemingly their prime.
It started well in the tournament, routing Slovakia, outlasting the host Russians in an eight-round shootout and cruising past Slovenia and the Czech Republic by a combined score of 20-6.
The Americans were humbled by defending champion Canada, which also beat them for the gold in 2010, with a 1-0 win that was more lopsided than the score indicated.
Then, they got embarrassed by Finland.
The United States seemed to bounce back from the crushing defeat to Canada with an inspired start, blocking shots with various parts of their bodies, including Phil Kessel's chest when he stood tall in an open net.
Selanne seemed to deflate the American' spirit with his backhander from the left circle that beat Quick's blocker for the first goal.
Before the Americans could catch their breath, the Finns took a 2-0 lead 1:38 into the second period when Jokinen caught Quick out of place after a cross-ice pass by shooting into an open net from the right side.
Finland won bronze for the second straight Olympics and third time since 1998, the first with NHL players. It lost to rival Sweden in the 2006 gold medal game and in Friday's semifinals.
The Finns were fired up for the consolation prize and weren't satisfied with a two-goal lead after two periods, pouring it on with Juuso Hietanen's goal 6:10 into the third — as a penalty on Kane was expiring — and the Finnish Flash scored for a second time less than two minutes later.
Selanne, the oldest player in the tournament, started the game with an Olympic record 41 points and extended his mark with a fantastic game.
Olli Maatta put Finland ahead 5-0 with 6:51 left in the game and U.S. coach Dan Bylsma left Quick in the game, leaving Miller staring expressionless across the ice from the bench.