Neither side could find a breakthrough in a tense round-of-16 tie that produced only a handful of clear openings, with Gregor Kobel and Camilo Vargas both required to make smart saves before extra time settled nothing either.
Colombia twice went close to snatching it late on, Jhon Lucumi heading against the crossbar from a corner before Jaminton Campaz fired over from a great chance after a Swiss defensive mix-up.
The shootout swung Switzerland's way when Davinson Sanchez crashed his effort against the bar and Cucho Hernandez saw his penalty saved by Kobel, leaving Ruben Vargas to fire home the winning kick and send his side through to face World Cup holders Argentina on Saturday.
Vargas, the match-winning penalty taker, admitted the achievement had yet to sink in.
"I don't think I've fully realized it yet. I thank God for this moment. I wasn't even sure if I would be able to play. We somehow managed to give it a try, and looking back now, I'm just grateful and happy that I was able to help the team. The team worked incredibly hard and fought for more than 120 minutes.
"It wasn't easy to play football here today. And now we've made history. It simply feels amazing."
Switzerland were missing breakout star Johan Manzambi, who picked up an injury in training the day before the game having contributed three goals and two assists across the tournament's opening four matches. Head coach Murat Yakin nonetheless got his selection and substitutions spot on, with his changes shaping both the closing stages and the shootout itself.
"I don't think you want to hear my match plan today, but it worked out exactly the way we wanted it to. And in the end, that's what matters," Yakin said. "It wasn't just about that, though. At the start, we needed experience. We needed the right mentality.
"Then, in the second half, we made a substitution that gave us even more control, especially in possession. And as the game went on, we were also able to bring on the players we wanted for the penalty shootout. You always have a plan. When it works out in the end, it's all the more satisfying. Of course, we also had a bit of luck today, and that's part of football."
It is the first time Switzerland have won back-to-back knockout matches at a single World Cup, and their first appearance in the last eight since they hosted the tournament in 1954. Colombia, meanwhile, are eliminated on penalties at the World Cup for the second tournament running, having also lost a shootout to England in the last 16 in 2018, and missed out on Qatar 2022 entirely.