The two sides were deadlocked 0-0 after 90 minutes and when another half an hour of extra time failed to produce a goal, the contest was destined to be decided by penalties. Lionel Messi stepped up to score Argentina's first spot kick but, as the first three Chileans converted unerringly, Gonzalo Higuain and Ever Banega missed.
That left the stage set for forward Sanchez, who made up for a quiet game by dinking the ball into Sergio Romero's net to send the capacity crowd at Santiago's National Stadium into raptures.
"Our whole idea was to win something, we knew that this was the moment," goalkeeper Claudio Bravo told reporters.
"This is a privileged generation and now it is time to enjoy it. We are not used to winning anything so this is an exception." Chile had beaten Argentina just once in 38 competitive matches before the final and were outsiders to take the trophy for the first time in the 99-year history of the tournament.
Few sides were able to handle their fast-paced football, however, and they won four of their five games on their way to the deciding match, scoring 13 goals and delighting their home fans.
"This is something so nice for all the Chilean people,” midfielder Arturo Vidal told reporters after the whistle.
"We Chileans needed a triumph, something like this Cup, today we took an incredible step, we're America's best." The final itself was a poor game with neither side able to create many chances.
With Messi uninspired and Sanchez faltering, the teams attempted just 19 shots between them -- less than all but one of the Copa America finals contested since 1993.
Higuain came closest to breaking the stalemate for Argentina in the last minute of regulation time.
Messi sent Ezequiel Lavezzi through on the leftand he crossed to Higuain at the far post but the striker could only put the ball in the side netting. Sanchez had one great chance to steal the game for Chile in extra time but blazed over with just the goalkeeper to beat.
(Writing by Andrew Downie, editing by Nick Mulvenney)
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