Alexis Mac Allister had given Argentina an early lead inside 10 minutes, only for Dan Ndoye to level for Switzerland in the 67th minute.
Five minutes later the game turned decisively when Breel Embolo was shown a second yellow card for simulation after a VAR review overturned an initial booking for Argentina's Leandro Paredes under the tournament's mistaken identity protocol, reducing the Swiss to 10 men.
Switzerland held out for the remainder of normal time, but Julian Alvarez produced a stunning long-range finish in the 112th minute before Lautaro Martinez added a third to complete the win.
Speaking after the match, Scaloni was candid in his assessment of a difficult afternoon for his side.
"The truth is we suffered today. We knew they were a very physical team. I think they put us in a lot of difficulty. We weren't able to get out of certain situations. 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.
"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," he said.
Julian Alvarez, whose moment of quality ultimately broke Swiss resistance, reflected on the difficulty of breaking down a side who spent much of the second half and all of extra time defending with 10 men.
"[I'm] very happy, above all. We kept trying until the end, things got difficult even though we had an extra man, the extra time... But well, we knew that if we kept going all together, the goal was going to come, and that's how it was.
"Obviously we would have preferred to win it earlier, but we know it's not easy. All the matches at the World Cup have been like that. And well, with two more to go, we're going to go all out for it," Alvarez said.
The victory sends Argentina through to face England in the semi-finals, with Scaloni's side now two wins away from becoming the first nation since Brazil in 1962 to retain the World Cup.
Switzerland, by contrast, were left to rue the officiating decision that shaped the contest, with head coach Murat Yakin left fuming afterwards and captain Granit Xhaka admitting the dismissal had changed the complexion of the game.
For Argentina, though, the result extends a remarkable run under Scaloni that has now delivered back-to-back World Cup semi-final appearances.