Alexis Sanchez netted the clinching goal from the spot after Gonzalo Higuain and Ever Banega had missed for Argentina.
The nail-biting finish came after a tense 0-0 draw in which neither side dominated and clear-cut chances were few and far between.
"This is something so nice for all the Chilean people,” midfielder Arturo Vidal told reporters.
"We Chileans needed a triumph, something like this Cup, today we took an incredible step, we’re (South) America’s best."
Sanchez dinked his penalty into the middle of the goal with Chile’s fourth kick to guarantee the shootout win.
It was Higuain who had come closest to breaking the stalemate for Argentina in the last minute of regulation time.
Messi sent Ezequiel Lavezzi through on the left and he crossed to Higuain at the far post but the striker put the ball in the side netting with Messi in a good position.
Sanchez had one great chance to steal the game for Chile in extra time but shot wide with just goalkeeper Sergio Romero to beat.