Arsenal's Alexis Sanchez slotted in the winning spot-kick after Argentina's Gonzalo Higuain fired over and Ever Banega was denied by Claudio Bravo.
Higuain had earlier missed from close range after 90 minutes of few chances.
Sanchez came closest in extra time for Chile, who scored all four of their penalties in the shootout.
Where Chile held their nerve, Argentina wilted in the shootout at the end of a performance that fell well short of the heights they reached in their 6-1 semi-final win over Paraguay.
Le Albiceleste are without a major trophy in the last 22 years since winning the 1993 Copa America in Ecuador.
Chile's victory brought an end to almost a century of competing in the Copa America without claiming a title.
They were one of four teams who played in the very first tournament in 1916 in Argentina but the best they had done until this year was runner-up, which they have achieved on four occasions (1955, 1956, 1979 and 1987).
The Estadio Nacional Julio Martinez Pradanos was a cauldron of noise on Saturday, full of flag-waving red-clad home supporters, some of whom had reportedly paid up to $25,000 for tickets.
They were not treated to a dazzling attacking display from their side but rather one of drive, discipline and, ultimately, composure, epitomised by the industrious and energetic Sanchez and Charles Aranguiz.
Both sides scored their opening penalty, with Matias Fernandez emphatically finding the top corner for Chile before Lionel Messi sent his kick low into the bottom corner.
However, after Arturo Vidal had powerfully dispatched Chile's second, Higuain blazed his effort way over the bar.
Charles Aranguiz made it 3-1 to the hosts and it remained that way as Ever Banega's tame effort was palmed away by the diving Claudio Bravo.
Sanchez showed supreme composure to chip his effort down the middle and in to win the game and the competition for his side.
Following Argentina's World Cup final defeat to Germany last summer, four-time Ballon d'Or winner Messi once again missed out on his first major international medal.
He put in a supreme showing in the Copa America semi-final demolition of Paraguay, playing a part in all six of his nation's goals, but on Saturday he was largely on the periphery throughout, with his most significant contribution being a trademark run towards the Chile box to set up a wasted chance for Higuain in the final minute.
The 28-year-old is not entirely to blame for his failure in Santiago. His team-mates did little to create the space the forward needs to be effective, not did they find him with the regularity he is used to when playing for his club.
Credit must also go to Chile, who were quick to close him down whenever he did receive the ball in a threatening position.
Napoli forward Higuain was criticised for missing a superb chance in the World Cup final defeat to Germany in Brazil last summer and he was once again in the spotlight for the wrong reasons on Saturday.
He was guilty of spurning an opportunity to win the game for his side in the last minute of normal time when he failed to turn in Ezequiel Lavezzi's low cross at the back post.
And his penalty miss - Argentina's first in the shoot-out - was that of a player whose confidence and composure has deserted him. It handed Chile an advantage they would not relinquish.
Chile midfielder Arturo Vidal: "This is a dream come true, this team deserves something like this.
"We left our life on the field. Chilean people needed this joy and we managed to give it to them.
"We'll fight a lot in the next World Cup. This generation of players deserved this."
Match ends, Chile 0(4), Argentina 0(1).
Penalty Shootout ends, Chile 0(4), Argentina 0(1).
Goal!  Chile 0(4), Argentina 0(1). Alexis Sánchez (Chile) converts the penalty with a right footed shot to the bottom left corner.
Penalty saved! Éver Banega (Argentina) fails to capitalise on this great opportunity,  right footed shot saved  in the bottom right corner.
Goal!  Chile 0(3), Argentina 0(1). Charles Aránguiz (Chile) converts the penalty with a right footed shot to the bottom right corner.
Penalty missed! Bad penalty by Gonzalo Higuaín (Argentina) right footed shot is too high. Gonzalo Higuaín should be disappointed.
Goal!  Chile 0(2), Argentina 0(1). Arturo Vidal (Chile) converts the penalty with a right footed shot to the bottom left corner.
Goal!  Chile 0(1), Argentina 0(1). Lionel Messi (Argentina) converts the penalty with a left footed shot to the bottom right corner.
Goal!  Chile 0(1), Argentina 0. Matías Fernández (Chile) converts the penalty with a right footed shot to the top left corner.
Penalty Shootout begins Chile 0, Argentina 0.
Second Half Extra Time ends, Chile 0, Argentina 0.
Corner,  Argentina. Conceded by Gary Medel.
Attempt blocked. Ezequiel Lavezzi (Argentina) right footed shot from the centre of the box is blocked. Assisted by Éver Banega with a cross.
Attempt blocked. Gonzalo Higuaín (Argentina) right footed shot from outside the box is blocked.
Foul by Charles Aránguiz (Chile).
Lionel Messi (Argentina) wins a free kick in the attacking half.
Foul by Marcelo Díaz (Chile).
Ezequiel Lavezzi (Argentina) wins a free kick in the defensive half.
Corner,  Chile. Conceded by Gonzalo Higuaín.
Matías Fernández (Chile) wins a free kick in the attacking half.
Foul by Martín Demichelis (Argentina).
Foul by Charles Aránguiz (Chile).
Lucas Biglia (Argentina) wins a free kick in the defensive half.
Foul by Jean Beausejour (Chile).
Lionel Messi (Argentina) wins a free kick in the attacking half.
Attempt missed. Charles Aránguiz (Chile) right footed shot from outside the box is high and wide to the right. Assisted by Arturo Vidal.
Alexis Sánchez (Chile) wins a free kick in the defensive half.
Foul by Lucas Biglia (Argentina).
Foul by Matías Fernández (Chile).
Lionel Messi (Argentina) wins a free kick on the right wing.
Delay over. They are ready to continue.
Delay in match Charles Aránguiz (Chile) because of an injury.
Offside, Chile. Matías Fernández tries a through ball, but Ángelo Henríquez is caught offside.
Attempt blocked. Marcelo Díaz (Chile) right footed shot from outside the box is blocked.
Corner,  Chile. Conceded by Marcos Rojo.
Foul by Arturo Vidal (Chile).
Nicolás Otamendi (Argentina) wins a free kick in the defensive half.
Second Half Extra Time begins Chile 0, Argentina 0.
First Half Extra Time ends, Chile 0, Argentina 0.
Attempt missed. Alexis Sánchez (Chile) right footed shot from the right side of the box is just a bit too high. Assisted by Claudio Bravo following a fast break.
Chile secured the first major trophy in their history on home soil courtesy of a penalty shootout win over Argentina in the Copa America final.