Chile had two players sent off and the game ended with angry protests by players and team officials aimed at referee Jan-Hendrik Salver.
The German official needed protection from stewards as he left the field, and then had to sprint into the tunnel as objects were hurled at him from angry Chilean fans.
The Argentines went ahead in the 12th minute with a fierce left-foot drive from Angel Di Maria, which was the first goal Chile had conceded in the tournament.
Chile's troubles mounted when Gary Medel was dismissed after he kicked out at Gabriel Mercado.
However, they almost grabbed an audacious equaliser when Gerardo Cortes's shot from the halfway line fell just wide.
After the break, Argentina suffered two blows, with Di Maria limping off with a muscle strain and Claudio Yacob's 63-minute booking, which will keep him out of Sunday's final.
But he doubled the lead two minutes later though Yacob, slotting home from close range after brilliant work from Maximiliano Moralez down the left.
Dagoberto Currimilla was then sent off -- apparently in response to his appeal for a red card for Yacob's foul -- prompting further protests with some substitutes joining the fracas on the field.
Moralez knocked-in the third in injury time, prompting further angry gesticulation from the Chilean players as the match ended in scenes of chaotic acrimony.