Alejandro Sabella's side was frustrated by a superbly resilient opposition until the Barcelona star bent home a fantastic strike in second-half injury time.
A sublime strike from Lionel Messi denied Iran a famous FIFA World Cup point as his late goal gave Argentina a 1-0 win in Belo Horizonte.
Few would have expected Argentina, viewed as one of the favorites to win the tournament, to be given too much of a test by a team that has never gone beyond the first round of a World Cup in three previous appearances.
However, Carlos Queiroz's side, which had been criticized for its defensive style of play following its opening Group F goalless stalemate against Nigeria, produced another resolute display at Estadio Mineirao on Saturday to come within a few minutes of a highly creditable draw.
But, in injury time, it was Messi who stepped up when it mattered most, curling home brilliantly into the corner to give the South Americans a place in the last 16.
Meanwhile, Iran will head into its final group game with Bosnia-Herzegovina on Wednesday reflecting on an opportunity missed, and knowing that only victory will be enough to extend its tournament.
Argentina unsurprisingly dictated the opening stages without creating any clear-cut opportunities against an Iran defense that displayed the same determination as against Nigeria.
Gonzalo Higuain was the first to test the Iran back line after 13 minutes, but goalkeeper Alireza Haghigi showed quick reactions to race out and block the Napoli striker's effort after he had been played through by Fernando Gago.
Higuain was involved again as he linked up with Sergio Aguero inside the area, only to see the Manchester City man's curling effort brilliantly kept out by a fine diving save from Haghigi.
Aguero then headed wide from a corner as Argentina's dominance of play continued, yet Iran, with impressive commitment and organization, did not have too much difficulty in keeping the South Americans at bay.PHOTOS: Beautiful people in Brazil | Brazil protests | Soccer fanatics | Fan tattoos
Iran's ability to maintain parity also owed much to Argentina's profligacy, which was demonstrated in the 35th minute when defender Ezequiel Garay headed a Messi free kick over the crossbar.
Queiroz's men were able to fashion some first-half opportunities of their own, the best of which came after 41 minutes as center back Jalal Hosseini turned Ashkan Dejagah's corner narrowly wide of the post.
Pablo Zabaleta went close from the edge of the area in the opening stages of the second half, however, it was Iran which continued to look more threatening.
Indeed, the team would have taken the lead in the 53rd minute had forward Reza Ghoochannejhad not directed his header straight at Argentina goalkeeper Sergio Romero following a fine break.
Iran felt it should have been awarded a penalty a minute later after Zabaleta appeared to bring down Dejagah, with replays suggesting that the South Americans had been given a let off.
At the other end, Messi failed to find a finish after surging forward with an archetypal run from midfield and curling wide of the post.
With time running out, Romero was called into action to tip Dejagah's header over and Iran had an even better chance when Ghoochannejhad was sent clear one-on-one but produced a weak finish.
But just as the Iranians looked set to celebrate a famous point, Messi struck.
The Barcelona man worked space on the edge of the area before opening his body to bend a shot into the corner, as Argentina's class – and more accurately that of the team's captain – finally told.