The South Koreans, semi-finalists at the 2002 FIFA World Cup, went into the match needing at least a draw to be sure of progressing and strikes from Lee Dong-Gook and Lee Keun-Ho saw them through as Group B winners.
Lebanon join Korea Republic in the draw for the next round, to be made in Kuala Lumpur on 9 March, after finishing second in the group despite suffering a surprise 4-2 defeat away to the United Arab Emirates.
Needing a win to advance, Kuwait started aggressively in Seoul, consistently breaking through with their quick short passing game, and the home side had goalkeeper Jung Sung-Ryong to thank for their clean sheet at half-time.
Kuwaiti midfielder Waleed Ali launched a shot from the edge of the penalty area in the seventh minute, forcing Jung into a punching save, before a Bader Al-Mutwa drive stung the keeper's palms around the half-hour mark.
But it was K-League star Lee Dong-Gook who broke the deadlock in the 65th minute, stroking home his fifth goal in four career games against Kuwait when the visitors failed to clear a loose ball. Six minutes later, Choi Hyo-Jin found Lee Keun-Ho in the box after intercapting a Kuwaiti clearance, and the forward made no mistake to give his side a two-goal cushion.
If Korea Republic make it through to Brazil 2014, it will be their eighth consecutive appearance at the FIFA World Cup and their ninth overall.