The match finished goalless after extra time before goalkeeper Gregor Kobel emerged as Switzerland’s hero, saving Cucho Hernandez’s penalty in the shootout to seal a historic victory.
Earlier in the shootout, Colombia centre-back Davinson Sanchez struck the underside of the crossbar, while Switzerland’s Manuel Akanji blasted his effort over, leaving the contest finely balanced.
Kobel’s stop ultimately settled the tie as Granit Xhaka, Zeki Amdouni, Cedric Itten and Ruben Vargas all converted their spot-kicks to send Murat Yakin’s side through.
The victory secured Switzerland’s first appearance in the World Cup quarter-finals since 1954, with the Swiss now preparing to face reigning champions Argentina at Kansas City Stadium on Sunday, 12 July.
“It was difficult in the beginning but we stayed in it and showed great mentality,” Switzerland coach Murat Yakin said.
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“Then I brought on the right players at the right moment. In the end it was important for the players who scored the penalties to be on the pitch then.
“It was just amazing, to see him smash that (Vargas penalty) in. I was so relieved and happy.
“This is a historic moment. We have reached the best ever for the Swiss team but the trip goes on.”
The match itself offered little excitement for long periods, with clear-cut chances proving scarce throughout normal time. Colombia came closest before the break when Jhon Lucumi headed a corner against the crossbar.
The South Americans threatened again during extra time, with substitute Jaminton Campaz forcing an awkward save from Kobel after trying his luck from distance.
Colombia then squandered a golden opportunity to settle the contest moments before penalties. Campaz curled an effort over the crossbar after Xhaka’s defensive error presented him with space, a miss that proved costly.
The defeat marked another painful World Cup exit for Colombia via a penalty shootout in the round of 16, repeating the heartbreak they suffered against England at the same stage of the 2018 tournament.