The match looked to be headed for extra time when Suarez blasted a curling shot in off the post to clinch an 80th-minute winner after South Korea had fought their way back into the game and could have won it.
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"Being young, this is the moment I have always dreamed of. This is an unrepeatable moment," said 23-year-old Suarez.
"As a football player you always want things to work out, even more so in the World Cup."
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Uruguay, who won the World Cup in 1930 and 1950, will play either the United States or Ghana in the quarter-finals in Johannesburg on July 2.
The defeat was a crushing blow for 2002 World Cup semi-finalists South Korea, whose players slumped to the ground in devastation after being denied a place in the last eight for the first time on foreign soil.
An early horrendous error by one of the tournament's leakiest defences left the Koreans chasing a game they went on to dominate, with Suarez popping up in clear space to slot the ball into an empty net in the eighth minute.
Edison Cavani neatly curled the ball out wide to Diego Forlan, whose incisive low cross was missed by the keeper and the sleeping Korean defence and reached the unmarked Suarez, who coolly side-footed home to spark ecstatic celebrations.
South Korea never gave in and piled on the pressure with a steady stream of dangerous counter-attacks, with Park Ji-sung linking up well with Lee Chung-yong and Park Chu-young, whose free-kick struck the post early on.
KOREAN ONSLAUGHT
The relentless Korean onslaught continued after the interval and they came close through Kim Jae-sung and Park Chu-young as Uruguay rarely threatened and looked to have settled for a slender victory.
South Korea almost levelled close to the hour when Park Ji-sung forced a save from Fernando Muslera, but their pressure paid off in the 68th minute when Lee Chung-yong out-jumped Muslera to head a poorly cleared free-kick into the net.
South Korea's celebrations were cut short by Suarez's clinical strike, but they kept on fighting and almost took the game into extra time in a thrilling last 10 minutes.
Lee Dong-guk's shot went through the keeper's legs but lacked the power to cross the line and the ball was hacked to safety to the despair of the Korean players.
"Our players never give up no matter what the situation," South Korea coach Huh Jung-moo said.
"We were on a high we were doing quite well ... the reason we lost was we were unable to capitalise on our opportunities."
Uruguay coach Oscar Tabarez admitted it had been a tough game but said they were delig
