Murat Yakin's team had impressed on route to their first-ever spot in the last eight, but they were eliminated by the holders, amid more controversy focused on Lionel Scaloni's side.
Switzerland were forced to play the final 15 minutes of normal time with 10 players, following Breel Emobolo's red card, with Julian Alvarez and Lautaro Martinez sealing the victory in extra time.
The call to dismiss Embolo came after a bizarre VAR review where referee João Pinheiro went to the pitchside monitor to check his previous decision to caution Argentina midfielder Leandro Paredes.
After the review, Pinheiro deemed Embolo to have dived, and reversed the booking for Paredes to show Switzerland's No.7 a second yellow card at the Kansas City Stadium.
Switzerland raged at the call, claiming the review was not needed, and only came about due to Pinheiro's original mistake.
"We were punished because of a mistake. There was no reason for that booking. I don’t understand it. It was an innocent situation; there was nothing malicious about it," Yakin stated after the game.
“That decision was simply unbelievable. I completely disagree with it. There was clear contact, I don’t understand how the referee and the VAR came to that conclusion.
“I wouldn’t say they were being favoured. We had a fair and open match. Both teams played football. Butt, football wasn’t the winner today. It was a decisive moment that determined the outcome of the match. We can complain now, but I have to congratulate Argentina."