The United States also advanced by beating the Czech Republic 7-0 in the quarter-finals, with John Gaudreau scoring a hat trick and Riley Barber adding two more goals.
The Americans will face Canada at Ufa Arena on Thursday, while Russia will play defending champion Sweden.
Alexander Khakhlachyov opened the scoring for Russia with a shot from the right circle on a power play at 6:07. Chrstoph Bertschy equalized with a shorthanded goal at 10:05 after stealing the puck from Russia's defence.
Mikhail Grigorenko restored Russia's lead, finding the empty net from Kucherov's pass at 29:07, but Martin Ness was quick to profit from another mistake by the host's defence to level at 2-2 less than three minutes later.
Sven Andrighetto, left unmarked in front of the Russia goal, scored from close range to put Switzerland ahead for the first time in the match at 48:57.
Kucherov equalized from a power play at 58:21 to send the game into overtime in which neither team scored.
Gridorenko and Kucherov scored for Russia in the second round of the penalty shootout, while Alessio Bertaggia scored with his first attempt but then failed to beat Andrei Vasilevsky, who stopped 41 shots.
Nyffeler made 33 saves for Switzerland, which has not lost a match in regulation at the tournament.
Gaudreau scored the lone goal of the first period when the Czechs were two men down at 11:45, and doubled the lead on a power play 28 seconds into the second period. Ryan Hartman made it 3-0 with a shot from the left circle 40 seconds later, and the Czechs substituted goalie Patrik Bartosak for Matej Machovsky.
Barber scored into an empty net on a power play at 28:44 and beat Machovsky in a one-on-one less than four minutes later.
Gaudreau, unmarked at the right, had his third goal on a power play at 39:47. JT Miller capped the scoring with a slap shot from the blue line, capitalizing on a two-man advantage early in the third period.
John Gibson made 31 saves for the victory.
"We played the game we needed to play tonight. You haven't seen our best hockey," United States captain Jake McCabe said. "The team that wins the tournament is the team that gets better every game. You have to improve every game. Playing Canada is going to be emotional again. We're going to be ready for it."
Canada edged the United States 2-1 in the preliminary round.
In the relegation round Wednesday, Finland beat Germany 8-0 with Ville Pokka and Aleksander Barkov scoring two goals each.