With some of the most dogged defending that has been witnessed in this edition, Iranians nearly pulled off a memorable draw before the 'Albiceleste' skipper produced a precision left-footer that curled past goalkeeper Alireza Haghighi, who had been brilliant under the bar.
With six points from two games, Argentina are well on course for a berth in the Round of 16 although their dependance on Messi and failure of the other forwards is becoming increasingly evident.
Iran predictably employed the ultra-defensive tactic by keeping atleast seven players in their own defensive third which frustrated the Argentine forwards Gonazalo Higuain and Sergio Aguero even though the South American side enjoyed 75 percent of the ball possession.
Messi operating from the attacking hole was always under double covering which kept him quiet save a few dead ball situations. His curling direct free-kick in the 32nd minute went over the horizontal.
The first time that Messi really could threaten was at the stroke of an hour when he sprinted down the middle and his left-footed placement using the instep was atleast a foot away from the near post.
He took a number of corner kicks and free-kicks but much to the dismay of the partisan fans failed to get the winner. Whenever Messi had the ball in possession, there was a trail of red-shirts following him. Defensive midfielder Andranik Timotian should be credited for keeping Messi quiet playing the role of a spoiler in the middle third.
With Diego Maradona watching from the stands, the 'Alibiceleste' couldn't effectively decode the chink in the Iran defence in the first 45 minutes.
The best chance for Argentina in the first half came in the 16th minute of the first half when Fernando Gago split the Irani defence with a lovely assist from the right which saw Higuain reach the spot but goalkeeper Alireza Haghighi showed good anticipatory skills to foil the attempt.
Haghighi again came to the fore when he effected a diving save off a curling placement from Sergio Aguero after Higuain set him up.
Ezequiel Garay and Marcos Rojo attempeted headers went off target as Iran didn't attack much apart from a few sporadic counters.
The closest they came was in the 53rd minute when their celebrated striker Reza Ghoochannejhad's straight header was saved by a diving Sergio Romero as the Iranian forward evaded Javier Mascherano to try the powerful nod.
Having gained confidence by keeping Argentina at bay, Iran started hitting the opposition defence on counters with Mascherano not exactly inspiring the highest level of confidence.
In the 66th minute Ashkan Dejagah nearly got Iran a goal but Romero spectacularly tipped it over crossbar denying the Fulham man what probably would have been his most memorable goal in the World Cup.
Argentina manager Alejandro Sabella ultimately changed both his frontmen Aguero and Higuain during the final quarter with Palacio and Ezequiel Lavezzi.
Palacio got into the act in the 83rd minute when his header was parried away by Haghighi.
In the 86th minute, Ghoochannejhad pouched on a counter ball and beat Mascherano for pace but his placement wasn't powerful enough with Romero effecting a diving save.