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.
|BBC World Service's Richard Padula in Santiago|
"It was a game in which the destroyers kicked, grappled and smothered the playmakers. Messi wasn't given a yard of space as Chile stifled Argentina's attack. In that regard Jorge Sampaoli's game-plan worked perfectly. Apparently he'd studied hours of tapes of Barcelona in action to discover the key to stopping Messi.
"But throughout the tournament the Chileans have shown a strong will-to-win, which at times could even be described as dirty. Jara's finger the perfect case in point. They out-fought Uruguay and stamped out Argentina's goal menace."
Chile make history
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).
|Chile end a century of hurt|
Chile won their first Copa America after 173 games played in this competition, dating back to the first tournament in 1916.
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.
How the shoot-out unfolded
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.
Alexis Sanchez scored the winning penalty in Santiago
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.
Lionel Messi failed to make an impact on the final
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.
More horror for Higuain
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.
Gonzalo Higuain missed Argentina's best chance in the 90th minute
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.
'We left our life on the field'
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."