Losses to Canada and Czech Republic in Group B had ruined Finland's good start to the tournament, and meant they finished in third place. That set up a meeting with the USA, who finished second in Group A and began this meeting in Saint Paul as favourites.
Though there was no separating the sides in the first period, the USA took just 35 seconds in period two to click into gear, as Cole Hutson opened the scoring.
It took Finland just over four minutes to draw level through Heikki Ruohonen, but a Too many men on the ice call midway through the match was costly, as Cole Eiserman struck on the resulting powerplay to ensure the Americans had a 2-1 lead going into the third period.
But two goals in under a minute turned the match on its head - Leo Tuuva was perfectly placed to divert a deflected shot into the net, before KalPa's Joona Saarelainen gave the Finns their first lead with a tidy finish from in front of the net with 06:22 left to play.
However, that proved to be too much time for Finland to hang on, and with just over a minute and a half to play, Ryker Lee's sharp one-timer tied things up again and sent the match to overtime.
That disappointment was put to one side by the Leijonat, who silenced the Grand Casino Arena crowd in the third minute of the extra period, when Matias Vanhanen found Valila on the right, and he powered the puck into the top left corner of the net to give Finland a 4-3 victory.
As the lowest-seeded team left in the competition, Finland will take on top seeds and neighbours Sweden in the Semi-finals at 23:30 Finnish time on Sunday 4th January.