The Manchester United player went off injured after just 28 minutes, while neither side could break the deadlock as the hosts eventually triumphed in the penalty shoot-out to lift the trophy for the first time.
Gonzalo Higuain missed a glorious opportunity to win it for Argentina, missing from point-blank range with the last kick in normal time.
And Di Maria told reporters after the game of his disappointment at falling short in the final, as they did in the World Cup last summer, and he also felt United team-mate should have been awarded a spot kick.
“Again it was decided in a matter of seconds, and again we could not achieve our objective,” he said.
“We did everything to win, we had many chances.
“Near the end, Pipita [Higuain] nearly got a goal, while Marquitos [Marcos Rojo] should have had a penalty awarded to him.
“Chile’s tactics in the match were perfect.”