Suarez delighted with double strike.
Uruguay took the lead through the Ajax man after eight minutes, but South Korea leveled matters in the second half through Lee-Young Chong.
But Suarez had the final say when he smashed in the winner 10 minutes before full time.
"I cannot explain my happiness, I have always dreamed about this moment," he told Uruguayan television after the game.
"Obviously you always want things to go well and to go far in the World Cup and we are happy.
"At 1-1 we were scared, they played well, but then they stepped off the gas and we became strong. I couldn't believe the second goal. All I want now is to celebrate. I dedicate this win to all the fans and my family."
Uruguay will face either the USA or Ghana in the next round.
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