That set up the tantalising prospect of two Europe vs North America showdowns in the last four.
The Swiss will now face Canada on 17 February, while Sweden will take on USA for a place in the final.
Beaten by Canada (5-0), the USA (9-0) and Finland (4-3) in their first three matches, before securing a narrow win against Latvia (1-0), Switzerland’s women pulled off a massive result against the host team in a must-win encounter that took place in front of a full house at the Shayba Arena.
Stefanie Marty opened the scoring for Switzerland in the 11th minute of the first period, before their goal was subjected to a bombardment from the Russians, in the second and third periods.
Swiss goaltender Florence Schelling produced some magnificent shot-stopping, as she recorded a total of 41 stops for her shut-out!
Then, with 20 seconds left before the final buzzer, the Swiss broke home hearts, doubling their lead on the counter with a goal from Lara Stadler, who rolled the puck into an empty net.
Russia will now face Japan on 16 February in the classification battle as they look to finish 5th in the final rankings. Canada take on Switzerland the following evening.
Earlier at the Shayba Arena, the other quarter final play-off saw Scandinavian neighbours Sweden and Finland engaged in a tense tussle.
Both teams harboured real hopes of progressing to the last four and the first two periods saw them fight out a predictably close-fought affair, before the match exploded into life in the last 20 minutes.
After the first period ended goalless, Venia Hovia opened the scoring for Finland in the 34th minute, before Anna Borgquist equalised just before the buzzer to end the second period.
The Swedes finally got the upper hand in the final period. Lina Wester took advantage of a goalmouth scramble to give her side a 2-1 lead, before the Finns levelled the scores once more through Emma Nuutinen.
However, the Swedes proved more clinical in the end, with first Emma Eliasson and then Emma Nordin finding the net to secure a 4-2 victory and set up a semi-final encounter with the USA at the Shayba Arena on 17 February.
The Finns will take on Germany a day earlier to determine their ranking in the final classifications.