The game then erupted in controversy on 36 minutes when both Messi and Medel were shown the red card after a VAR check following an encounter between the pair.
"I didn't want to be part of this corruption, we shouldn't have to be part of this disrespect we suffered during the Copa America".
The expulsion marked the first time the Barcelona forward had been sent off in a senior game for club or country since his debut with the national team in 2005.
"We leave this Cup with a triumph but above all with our heads up and with the feeling that this time football was not fair with us", Messi said in an Instagram post.
These two teams will next play on September 5 in Los Angeles in a friendly in what's their next scheduled match.
Much like the semi final against Brazil, this match had controversy.
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Interaction with locals can be limited at times, but I can shop and dine nearly anywhere I want", he wrote. While at ANU, Mr Sigley says he "developed a strong interest in North Korean film and literature".
He suggested the red card had been punishment for those comments, saying: "With a yellow card it would have been over for both of us, but, well, maybe I'm paying for what I said last time".
Tampers began to flare as both sides saw red 37 minutes gone in Messi and Gary Medel, who got embroiled in a tug-of-war.
There were complaints from both teams, and several players continued arguing with each other.
Messi continued to be vocal post-match, however, the five-time Ballon d'Or victor believes the future is bright for Argentina.
Messi added that he fully expects hosts Brazil to beat Peru in the final on Sunday in Rio de Janeiro to lift their first Copa crown since 2007.
Argentina's third-place finish will likely prove little consolation for Messi and his team-mates, who lost to Chile on penalties in the previous two Copa America finals in 2015 and 2016.