The Group A game was held up for almost 10 minutes in the first half after angry fans hurled projectiles in protest at a penalty awarded to South Korea.
Stones and bottles rained down after striker Lee Dong-gook scored from the spot to give the Koreans a 2-0 lead in the 28th minute.
Teenage forward Park Chu-young had put the visitors in front in the 16th minute in sweltering Kuwait City.
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South Korea have qualified for every World Cup since 1986 and reached the semi-finals as co-hosts of the 2002 finals.
A lacklustre draw in Uzbekistan last Friday left them needing to win in Kuwait to be sure of extending their proud World Cup record.
"We promised we would get the four points we needed and we have delivered," said South Korea's Dutch coach Jo Bonfrere. "I salute the players. It's a young team and they are just going to get better."
Earlier on Wednesday Japan and Iran booked their places at next year's finals alongside hosts Germany.
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