New Zealand's match-winner came from Andy Hayward, who converted their first penalty corner by beating Argentine goalkeeper Juan Vivaldi with a low shot.
This was the fifth penalty corner conversion for Hayward, who played a key role in New Zealand's campaign that saw them top the round-robin league after posting victories in their initial four matches.
New Zealand have in past editions of the Azlan Shah Cup claimed the bronze medal four times, including consecutive podium finishes in 2008 and 2009.
World number seven New Zealand were the third highest ranked team in the tournament, behind Great Britain (number four) and South Korea (number seven).
Argentina did well to make it to the final after losing their first two matches, but were not able to breach the New Zealand defence led by goalkeeper Kyle Pontifex.