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Gregor Kobel was Switzerland's penalty hero, pulling off a stunning shootout save against Colombia's Cucho Hernandez to set up a quarter-final clash with Argentina.
The keeper pushed away Hernandez's spot-kick after the 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 on 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 in Vancouver.
Murat Yakin's side have reached the last eight of the World Cup for the first time since 1954 and they 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 clear 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 mishap by 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's lack of clinical finisher costs them
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 they 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 the focal point of a striker.
Campaz and Juan Quintero were lively off the bench but the former's missed chance late on 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 tournament's breakthrough stars, Johan Manzambi. He missed out through injury, and Switzerland missed his attacking verve and pace, which has seen him score three goals and create two more in this tournament.
Vargas, who had scored two goals earlier in the tournament, was also only fit for the bench but came on late and scored the winning spot kick. They will hope both can be fit for Sunday's quarter-final.
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