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.
BBC