It's the fifth time in the last six Winter Games that Finland have progressed to at least the semi-final stage.
An 11-0 thumping of Italy in their final group match secured Finland's place as the best runner-up in the groups and a direct ticket to the quarter-finals, while the Swiss had reached this stage by putting three unanswered past the Italians in the qualification playoffs.
But the extra rest didn't appear to be doing the Finns any favours - just over three minutes in, Damien Riat was called for interference, but not only did Finland's poor powerplay record extend to 12 failures to score from 14 across the tournament, but they didn't even generate a shot on goal.
The shot tally was only 3-2 in Switzerland's favour as a cagey first period entered its final six minutes, at which point Finland goaltender Juuse Saros inadvertently helped his opponents find some attacking form.
Saros was far too casual controlling the puck behind his goal, coughing up possession to Ken Jager, and the Nashville Predators' goalie couldn't get back in position in time to stop Riat applying an easy finish from Jager's pass.
Just over a minute later, the Swiss completed a quickfire double to go 2-0 ahead - Philipp Kurashev maintained possession well behind the net and found Pius Suter, who laid the puck back for the oncoming Nino Niederreiter to blast first-time past Saros before he could react.
In the second period, the Leijonat showed much more attacking intent, but could not find a way past the impressive Leonardo Genoni in the Swiss goal, as he made saves from Sebastian Aho, Miro Heiskanen and Mikko Rantanen to name a few.
At the other end, Saros atoned for his earlier error midway through the match, making a huge save to deny Nico Hischier from point-blank range to keep his team in the contest.
Genoni took his save tally to 20 by the 40-minute mark, yet his teammates failed to capitalise on their first powerplay of the match late on, after Artturi Lehkonen was called for tripping, and the second period ended scoreless.
A third-period fightback from Antti Pennanen's side was struggling for lift-off as Eetu Luostarinen was sent to the box for boarding just over a minute in. They survived the powerplay before getting their own man-advantage in the eighth minute, after Janis Moser was penalised for interference.
But their best chance in those two minutes came and went after Heiskanen went one-on-one with Genoni, but failed to get the shot away.
However, the Finns finally found a way past Genoni with 6:06 left in the game, as Luostarinen played the puck across the ice from the left to Aho, who shuffled inside from the right and fired a clean effort past the goaltender to set up a grandstand finish.
Roope Hintz and Mikael Granlund both were denied by Genoni with time winding down, forcing Finland to remove the goaltender, and with 1:12 left, the extra attacker came to Finland's rescue.
Rantanen on the right fed the oncoming Hintz through the middle, and he quickly shovelled the puck onto Heiskanen on the left. His shot was going wide, but a deflection off Jonas Siegenthaler's stick carried it in for 2-2, sending the match to overtime.
Switzerland could have won it had Sven Andrighetto made true connection with a pass in front early in the extra period, but in the end, it was the Finnish momentum that carried them over the line.
Anton Lundell broke out of his own zone and played the puck down the middle for Lehkonen, who got in behind the Swiss defence and kept his composure when facing Genoni, and slotted home the winning goal with 03:23 played.
Finland march on and join Slovakia and Canada in the semi-final line-up, with Sweden facing the United States later this evening for the fourth and final place.
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