The Czech team appeared en route to victory before Switzerland scored twice in the last 11 minutes of regulation, forcing overtime, and Wey thrust her fist in the air after whipping the puck past the Czech goalie for the Group A shootout win.
A nifty solo effort from 19-year-old forward Tereza Plosova provided a two-goal cushion for a Czech team eager to bounce back from their 5-1 loss to the United States on Thursday, but they were unable to hold off a Swiss comeback.
Switzerland next play defending Olympic champions Canada on Saturday, while Czech Republic play Finland on Sunday. All five sides from Group A will advance to the last eight.
The two sides traded goals in rapid succession in an exciting first period, as forward Kristyna Kaltounkova slipped the puck through the legs of the diving Swiss goalie less than 90 seconds into the game.
Swiss forward Laura Zimmermann levelled it in the eighth minute, but Czech Republic regained the lead more than a minute later, when Natalie Mlynkova flipped in an easy wrist shot from close range off a fine assist from Katerina Mrazova.
Goaltender Michaela Hesova pounced on the puck to keep the Czech lead late in the scoreless second period, before Plosova made it 3-1 just over six minutes into the third.
But Switzerland fought back as Alina Muller scored on the power play 10 minutes into the period and defender Lara Christen rifled the puck into the goal with less than three minutes left in regulation, pushing the game into overtime and the eventual shootout.
"We always believed that we can play over 60 minutes, and that's what we did. That's why we came back and won that game," said Zimmermann.
"It's always important to come back, even if you're nervous, just play your best game, and that's it. We never stopped believing."
France falls again
France, making its first Olympics appearance in women's ice hockey, came into the match at the Milano Rho Arena already under pressure after losing 4-1 to hosts Italy in their opening game on Thursday, and a goal in the dying seconds was too little, too late against Japan.
Rui Ukita put Japan ahead late in the second period but France tied it just over a minute later through Lore Baudrit.
Japan's Makoto Ito and Suzuka Maeda scored in quick succession late in the final period before Gabrielle De Serres scored for France.
Japan tested France goaltender Alice Philbert early on, firing six shots on goal in the opening six minutes, and outpacing the French, who struggled to get beyond their own blue line.
Japan opened the scoring in the 19th minute of the second period, when Ukita jumped from the bench, collected the puck as soon as her skates hit the ice and fired past Philbert from the right face-off circle.
France took just over a minute to draw level, with Baudrit converting during a scramble in the crease. Japan failed to take advantage of a power play early in the final period but went ahead through Ito with less than four minutes left.
Late drama
The French pulled their goalie for an extra skater but Japan made them pay. Maeda managed to get her stick to the puck in the neutral zone to send it goalwards, and her effort found the empty net despite a desperate dive by France forward Chloe Aurard-Bushee.
A late power play gave France hope and, after again pulling Philbert, De Serres scored from long range with 13 seconds left on the clock, but Japan hung on to avoid overtime.
The top three in Group B make the quarter-finals, where Sweden began with a 4-1 win over Germany on Thursday.
The two games were the only contests on a lighter Friday ice hockey programme, with the evening session cleared for the Opening Ceremony, which officially kicks off the Games.