Chile took the early lead, going ahead after only 11 minutes when Felipe Gutierrez leapt above the visiting defence to head past Sergio Romero.
As if sparked into action by falling behind, and seeking to reply to the fantastic support of the visiting supporters, Argentina responded quickly.
They were level after eight minutes, when Di Maria finished smartly after some excellent build-up from Ever Banega.
Six minutes later, the South American giants took the lead through Gabriel Mercado.
La Roja failed to clear their lines, and as Messi pressured the defence, the ball flicked up to the River Plate defender, who finished acrobatically past Claudio Bravo to put Gerardo Martino’s side into the lead.
Chile pressed for an equaliser during a tetchy second half, with five yellow cards shared between the two sides.
However, weakened by early injuries to Matias Fernandez and Marcelo Diaz, they were unable to rediscover their early vigour and fell to a disappointing defeat.
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Victory takes Argentina up to fourth on eight points, one behind second-place Uruguay and five behind surprise leaders Ecuador.
Chile remain in sixth on seven points, level with Brazil who host Uruguay in a showpiece bout on Saturday.