Messi blazed a penalty over the bar in the shoot-out before Lucas Biglia missed to leave Chile's Francisco Silva with the task of stroking home the winning spot-kick at the MetLife Stadium.
It was a carbon copy of last year's Copa America final, also won by Chile on penalties after a goalless draw. Argentina's players meanwhile were left disconsolate as they digested a third defeat in a major final following losses to Germany in the 2014 World Cup and Chile in last year's Copa.
The defeat also ensured Argentina's 23-year wait for a major title - and a first for five-time world player of the year Messi - continued. The final had gone to penalties after finishing deadlocked at 0-0 after 120 minutes.
Both sides were reduced to 10 men in the first half as Brazilian referee Heber Lopes struggled to get a grip on an ill-tempered contest played in front of a crowd of 82,026.
The first yellow card of the match came in the 16th minute when Marcelo Diaz brought down Messi with a cynical hack across the thighs as the Barcelona star advanced on the Chilean goal.
Argentina should have taken the lead on 21 minutes when a defensive blunder from Gary Medel, miscontrolling a back pass, allowed Gonzalo Higuain to go clean through on goal.
Higuain had time and space but opted to lift his shot over the advancing Bravo and it rolled wide of the post in a huge let-off for Chile.
It was a nightmarish case of deja vu for Higuain, who missed potentially match winning chances in both the 2014 World Cup final and last year's Copa America.
Seven minutes later Chile were left rocking when Messi burst forward only to be bodychecked blatantly by Diaz.
For a moment it appeared that referee Lopes had forgotten Diaz's earlier caution but eventually he brandished a red card and Chile were down to 10.
It ought to have been a warning to both sides but the spiteful tone continued for the remainder of the half, with Javier Mascherano and Arturo Vidal next into the book in the 37th minute.