The two players then aggressively bumped chests a number of times before the match referee intervened and immediately flashed reds for both.
Incredibly, Paraguayan referee Mario Diaz de Vivar showed the two team captains the red card.
This was Messi's first red card with Argentina since 2005.
Perhaps that was an omen of how his Albiceleste career would go, as Messi has suffered disappointment after disappointment during his quest for an worldwide trophy.
In stoppage time, the Chile bench were all up claiming for handball, but this time there was no extensive VAR review as the appeals were swiftly waved away.
With Brazil set to face Peru in Sunday's decider, Messi added: "Hopefully, the referees and VAR won't influence it and they'll let Peru compete, but that looks hard to me". The match with "pentakampeonami" could not be held at the highest level - the hosts of the tournament won 2-0 and went to the final.More news: Federer crushes Clarke to reach Wimbledon third round
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Much like the semi final against Brazil, this match had controversy. "A yellow would have ended it for both of us". With both teams surely disappointed after missing out on the final, that didn't dampen their intensity in what was a physical, heated and at times dirty battle. But moments later, Messi and Medel walked off the field.
Messi said: "I always tell the truth".
There was no let-up in the intensity after the break and Chile gained a foothold in the game via a 59 minute penalty, awarded via another VAR check, which was converted by Arturo Vidal.
Dybala justified his starting berth by doubling Argentina's lead 10 minutes later with a neat finish over Arias after being set up by Giovani Lo Celso.
Argentina's failure to reach the final left Scaloni under pressure and his contract is up at the end of the year, but avoiding defeat here will have taken some of the heat of the Albiceleste coach.