Elijah Just gave New Zealand an early lead in the seventh minute after being served by captain Chris Wood.
Iranian captain Mehdi Taremi hit the post in the 23rd but his team was level in the 32nd after Ramin Rezaeian collected a loose ball and hit it into the net.
Iran thought they had taken the lead before the break but Ali Nemati's goal was ruled out for offside.
It was New Zealand, however, who were first on target in the second half with Just linking up once more with Wood to complete his brace.
But the Iranians didn't waste too much time and equalised again thanks to Mohammad Mohebbi in the 64th.
Both sides kept the pressure up and it was an open game until the last minute, but the teams eventually had to settle for a point each.
New Zealand are competing only in their third World Cup, first one since 2010, and are still seeking a first win in the tournament.
Iran were one of the first teams to qualify for the event in the United States, Mexico and Canada but the nation's conflict with the US and Israel cast doubt over their participation at the tournament
At some point it was even unclear whether Iran would be permitted to play their three group-stage matches in the US. Amid ongoing issues over visas and entry to the US, the national team moved their base from Arizona to Mexico to avoid further problems.
New Zealand top Group G standings ahead of Iran but they are both level on one point with Belgium and Egypt, who played out a 1-1 draw earlier.
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