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Gregor Kobel was Switzerland's penalty hero as he pulled off a stunning shootout save from Colombia's Cucho Hernandez to help set up a quarter-final against Argentina.
The keeper pushed away Hernandez's spot kick after centre-backs from both sides had earlier missed.
Colombia's Davinson Sanchez rattled the underside of the bar but former Manchester City defender Manuel Akanji skied his spot-kick to restore the balance.
But Kobel's stop from Hernandez was impressive before Cedric Itten and Ruben Vargas scored to add to earlier successful penalties taken by Granit Xhaka and Zeki Amdouni as Switzerland won the shootout 4-3 in the last-16 tie at BC Place Vancouver.
Murat Yakin's side have reached the last eight of the World Cup for the first time since 1954 and will face reigning champions Argentina at Kansas City Stadium on Sunday, 12 July at 02:00 BST.
Extra time and the penalty shootout followed a rather uneventful 90 minutes in which both sides struggled to create many great chances.
Jhon Lucumi went close for Colombia but his header from a corner struck the crossbar, before a long-range effort from substitute Jaminton Campaz forced an unorthodox save from the Swiss keeper as the game threatened to burst into life in the first half of extra time.
Colombia should have won it late in extra time but Campaz curled a shot over after a defensive mistake from Xhaka.
It was a miss they would live to regret as the South Americans suffered shootout heartbreak again after losing to England in 2018 on spot-kicks at this stage.
Colombia exit having conceded just one goal in the tournament. They topped their group ahead of Portugal to give them a more favourable path in the knockout stages and saw off Ghana in the last 32.
But here they couldn't find a breakthrough under the roof in a sweltering stadium predominantly filled with boisterous Colombian fans.
Luis Diaz, who did score from the spot in the shootout, and James Rodriguez, who became their record appearance maker with 131 caps, tried to create chances in the first half but they struggled without a focal point.
Campaz and Juan Quintero were lively off the bench but the former's missed chance late in extra time meant Colombia did not add to their tally of five total goals from 94 shots.
Switzerland have now won two knockout games in the World Cup for the first time, having beaten Algeria in the last 32, and now have a chance to stop Lionel Messi and Argentina's run to back-to-back World Cup glory.
They progressed here without one of the breakthrough stars of the tournament in Johan Manzambi. He missed out through injury, and Switzerland missed the attacking verve and pace which has helped him score three goals and create two more in this tournament.
Vargas, who scored two goals earlier in the tournament, was also only fit for the bench but did come on late and scored the winning spot-kick. They will hope both can be fit for Sunday's quarter-final.
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