"It's the first time we are reaching the second round away from home," Huh Jung-moo, South Korea's coach, said.
"That was our goal. I'm very proud of my players. I feel that my players have
played to their full potential."
Nigeria blew a great chance to be Africa's only representative in the second round of this year's competition, as with Greece loosing 2-0 their destiny was in their own hands.
Argentina, who were captained by Lionel Messi, at 22 the youngest ever captain for the country, will play Mexico in the round of 16. South Korea will play Uruguay.
Earlier in on Tuesday, Uruguay beat Mexico 1-0 to finish top of Group A.
Luis Suarez scored the winner in the 43rd minute, while Mexico still advanced to the second round thanks to goal difference.
In the other final game of Group A, France were put out of their world cup misery with a 2-1 loss to South Africa.
Despite going out of the tournament, hosts South Africa gained some solace in their first victory of the tournament.
For France it ends a disastrous campaign, after striker Nicolas Anelka was sent home for swearing at coach Reymond Domenech last week and the squad subsequently refused to undertake a training session.
They finished the game with 10 men with Yoann Gourcuff being dubiously sent off for an elbow on Macbeth Sibaya.
France had already gone 2-0 by the 37th minute with goals for South Africa from Bongani Khumalo and Katlego Mphela. Florent Malouda pulled one back for France in the 70th minute.