The 1-0 win meant South Korea qualified for the semi-finals for the second successive tournament.
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The Flying Eagles, who eliminated hosts and six-time champions Argentina in the Round of 16 on Wednesday, created few chances despite passing the ball around the field quite well.
They were clearly lacking the necessary punch in the final third.
Victor Eletu, whose show of guts and grits led to the second goal that ended Argentina’s hope on Wednesday, curled a fine shot from outside the penalty box on the half-hour.
But the Korean goalkeeper punched it out of danger.
Both teams watched each other closely in a highly cagey affair, and only a few risks were taken throughout the first period.
In the second period, Nigeria looked more adventurous, with defender Daniel Bameyi striking from the edge of the penalty box, only to see his effort stopped by the goalkeeper.
In added time, Emmanuel Umeh’s dazzling run ended with a shot also smartly held by the Korean goalkeeper.
The South Koreans got the winner five minutes into the first period of extra time when substitute Sun Choi met a corner kick with a glancing header.
Umeh, with a great chance to equalise in the 109th minute, blazed over the bar and, in the final minute, struck the side of the net.
The Flying Eagles will now return to Nigeria on Tuesday.
South Korea will face Italy in the semi-finals on Thursday.
The competition’s first quarter-final matches took place on Saturday, where Israel eliminated Brazil 3-2, and Italy ended Colombia’s dream 3-1.