Switzerland looked up to the challenge of the defending champions in the first 10 minutes of the match in Kansas City, dominating possession but unable to find a way through the Argentinian defence.
Their defensive efforts were rewarded in the 10th minute when Alexis Mac Allister headed home a Lionel Messi corner.
Switzerland continued to try to generate something dangerous, but were unable to find the decisive ball they needed.
It was not until the 44th minute that they finally found a decent opportunity, receiving a free kick on the left side of the box.
However, Remo Freuler pushed his attempt too far and the ball sailed harmlessly over the goal line.
The Swiss continued building pressure into the second half, finally drawing some activity from Argentinian goalkeeper Emi Martinez in the 60th minute, when a diving save was needed to stop a header from Breel Embolo.
Martinez was suddenly at the centre of the action, batting away another header from Dan Ndoye, before repelling a booming shot from deep from Granit Xhaka.
Switzerland's ascendancy finally paid dividends in the 67th minute, Ndoye on the receiving end of a nice one-two with Ricardo Rodriguez, generating a tiny bit of space for himself and finding the far corner of the goal.
However, their momentum came to a screeching halt just after the 70-minute mark.
What was initially given as a yellow card to Leandro Paredes for a challenge on Embolo was reversed by VAR and instead Embolo was booked for a dive - his second yellow of the match.
Switzerland were suddenly down a man and Argentina promptly took charge, camping in their attacking third and probing for a breakthrough.
Messi, who had been kept relatively quiet, looked likely to be the man for Argentina once again when he found space at the top of the box, but he was unable to curl his shot inside the right post.
Switzerland appeared to be content to push the result to penalties as extra-time began, sitting deep and allowing Argentina to explore for a winner.
They were forced to endure multiple Argentinian chances, including a Messi free kick in the 100th minute that flew into the wall, while rarely braving an attack of their own.
But their defensive stubbornness finally ended in the 112th minute, Julian Alvarez delivering a gorgeous strike from outside the box that dipped just under the top-right corner of the crossbar.
Lautaro Martinez ultimately sealed the win with a breakaway goal at the start of added time.
After the match, both coaches understandably had different views of Embolo's send-off.
"The truth is that luck was on our side today, because they had a player sent off and, from there, the team went on the attack,'' Argentina coach Lionel Scaloni told fifa.com.
"We have to be realistic, we have things to improve on, but winning is always better. That said, what this team achieved today is historic. Even though we could have played better, being in a semi-final again is historic."
Switzerland coach Murat Yakin expressed his disappointment at the decision.
"We were punished because of a rule that in my opinion is completely unacceptable," Yakin said.
"I don't understand. It's very painful that we were eliminated that way. I don't think we deserve that today, in my opinion.
"My boys are the real heroes. They put all their heart and their passion into their performance. I am very proud."
- with AP
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