Snitil was placed second after the round-robin stage but saw his bid for a medal dented twice in quick succession as he lost the page one-two play-off to Sweden and then the semi-final to Norway.
That saw Snitil go from potential gold medallist to one last-gasp match for bronze against Denmark's Rasmus Stjerne, who lost the page three-four play-off to Norwegian Thomas Ulsrud.
However the Czech Republic skip was nOt daunted by the situation and kept his cool to give his rink a 6-2 lead at the half-time break with a hit for three points in the fifth.
Stjerne responded with a hit for two in the next end only for Snitil to stretch the lead again after drawing for three in the seventh before Denmark conceded in the eighth with the score at 12-4.
Bronze marked the Czech Republic's first medal at a major international curling event and an emotional Snitil struggled to hide his emotion at what he achieved in Karlstad.
"Unbelievable, I'm almost crying, I'm so happy. It was a tight game. I think we figured out the sheet of ice better than the Danish, which I think decided the game," said Snitil.
"This is the first medal for Czech Republic men or women, the first big one. It is very satisfying for Czech curling. We have been working hard; we have a good junior programme.
"We have a newish club, so everyone has been working hard to achieve results. This is a good day for the association, a big day."
Meanwhile with the women's semi-final also taking place Russian Anna Sidorova prevented the host nation Sweden from reaching both finals in Karlstad.
With Swedish men's skip Niklas Edin safely in that final, Margaretha Sigfridsson would have been desperate to follow suit however she could nOt navigate past Sidorova.
Sidorova led 3-2 in the second end before Sweden swung the advantage their way 5-4 in the fifth however an extra end was to be required with the score locked at 6-6 after ten.
Russia were to run out 8-6 winners to book a place in the final against Eve Muirhead while Sigfridsson, who lost to the Scot in the page one-two play-off, will battle for bronze against Denmark.
Muirhead not only has the general momentum going into the final having won five straight in Karlstad but the psychological advantage too after beating Sidorova 6-5 in the last match of the round robin.
Although Sidorova is determined to ensure that counts for nothing, she said: "I remember our earlier game against Scotland. It was really tight and they only won in the last end.
"The percentages of my team were not as good as they could be. So we have a bit more confidence now that we have medals and I'm sure we can play even better."