In a repeat of last year’s clash, La Roja emerged victorious on penalties to condemn Gerardo Martino’s side to their third consecutive final loss.
Nicolas Otamendi started and Sergio Aguero came off the bench in a goalless 120 minutes of football.
Both sides missed their first spot kick, as Arturo Vidal’s was saved and Lionel Messi blazed over the bar.
The following five efforts, including Kun’s, were successful before Claudio Bravo thwarted Lucas Biglia, allowing Francicso Silva to bag the decisive strike.
The disappointment led to Messi vowing to quit international football, while Aguero also hinted more players could follow suit.
He told press: ”There are various players who are evaluating not to continue with the national team.
“It is the worst dressing room I’ve ever been involved in. Worse than the World Cup final in Brazil and the other Copa America.”
Argentina had started brightly, creating six chances in their first 25 minutes, including a header by Otamendi which flew just wide.
The defender was named in the tournament’s best XI after the game.
Both sides were reduced to ten men in the opening 45 minutes, with Marcelo Diaz and Marcos Rojo sent off.
Kun was introduced on 70 minutes and almost made an impact, spurning a couple of opportunities before forcing Bravo into an excellent stop - deemed the save of the competition - in extra time.
