The attacking midfielder, one of seven changes made by head coach Uli Stielike for the Group A clash, darted to the near post to meet Cha Du-ri's right wing cross in the 36th minute to break the deadlock.
It was a rare chance fashioned by the Koreans, who dominated possession at the Canberra Stadium but struggled to unlock the Kuwait deep backline and were guilty of lumping aimless long balls forward.
Injury and illness had forced Stielike to name an all changed front four from the side that beat Oman 1-0 in their opener and the replacements struggled.
One of them, former Asian Football Confederation Player of the Year Lee Keun-ho, wasted Korea's best, and only real chance six minutes before Nam broke the deadlock.
The 29-year-old was sent clear in behind the Kuwaitis after a through ball by Kim Min-woo but the striker's shot was easily pushed over the crossbar by goalkeeper Hameed Youssef.
Kuwait offered little going forward, a couple of crosses into the penalty area the only moment of concern for Korea's replacement goalkeeper Kim Seung-gyu.
Kuwait coach Nabil Maaloul had made four changes to his side from their opening 4-1 defeat by hosts Australia, with at least a point required to keep alive their slim knockout stage hopes.
Australia face Oman later on Tuesday in the other Group A match.
(Writing by Patrick Johnston; Editing by Sudipto Ganguly)
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