The defending champions turned the game from 2-0 and 3-2 down in a replay of last year's final, then behind 4-3, but a goal by Lukas Sedlak in the 57th minute sent the game into overtime.
The game was then decided at 63:30 by captain Roman Cervenka, who scored.
The Swiss had a much better start to the game, taking the lead after just 94 seconds when Christian Marti's shot from the corner caught Vejmelka short, who didn't shut the door above his right shoulder.
The Swiss continued their momentum in the sixth minute when they got the puck behind the Czech goalie for the second time.
Sandro Schmid's joy was short-lived, however, as the Czech team challenged and it was revealed the goal was preceded by an offside.
The first big opportunity for the Czech team came in the 18th minute, when Ondrej Beránek had a breakaway, but due to a foul he was unable to finish, resulting in a penalty shot, which David Pastrnak then missed.
Less than a minute later, Damien Riat found the rebound and gave the Swiss a two-goal lead. At the end of the first period, Radim Rulík's team got the advantage of a 5-on-3 power play, which they capitalised on thanks to an accurate shot by Matej Stranský.
The Czech team entered the second period with more verve, but Leonardo Genoni in the Swiss goal covered all the dangerous attempts.
At the beginning of the 27th minute, he capitulated when he was not able to save Filip Zadina's shot, which tied the score at 2-2.
The Czechs dominated the ice after the equaliser and turned the huge pressure into the leading goal, which was scored by defenseman Filip Pyrochta in the 36th minute with a shot from the blue line.
The third goal was also scored by Daniel Vozenílek, who made a great screen in front of Genoni.
Although the end of the second period belonged to the Czech Republic, the entry into the third act was better managed by the Swiss.
After two minutes, they managed to take advantage of a power play thanks to a beautiful combination at the end of which Schmid was on top. But that was not all for the defending silver medalists.
In the 49th minute, Sven Andrighetto returned the lead to the Swiss, who took advantage of a slight hesitation of Pastrnak, skated between the circles and shot a shot through the helpless Vejmelka.
The Czech team finally managed to equalise in the 57th minute, and for the second time in the game, it was during a power play. Lukas Sedlak showed how strong he can be in the crease and shot the puck into the net from the air.
The winner of the game had to be decided in overtime. Roman Cervenka scored the winning goal in the 63rd minute, taking advantage of a great pass from Pastrnak.