A match between two teams who will feature in January's Asian Cup was settled when Azmoun found the net with eight minutes to play in Tehran.
The 19-year-old Rubin Kazan forward pounced to seal victory for Carlos Queiroz's side after Javad Nekounam's free-kick had hit both posts.
Iran were therefore able to celebrate a first win since June, beginning their rebuilding process after a winless World Cup, while an inconsistent South Korea have lost three of their last six.
The visitors had the better of the opening half, with Bayer Leverkusen forward Son Heung-min going closest, only to be denied by Alireza Haghighi in the Iran goal.
A goalless draw appeared likely as the game drifted towards its conclusion, but the decisive moment arrived in the 82nd minute.
Nekounam's curling set-piece beat Kim Jin-Hyeon and, although the goalkeeper was twice saved by the woodwork, there was nothing the 27-year-old could do as Azmoun headed home the loose ball from close range.
Iran have now claimed three successive 1-0 wins over South Korea and the two sides could yet meet again in the Asian Cup semi-finals.