Suarez, who had five shots on target, scored the winner in the 80th minute after sidestepping two defenders and curling a right-foot shot off the post from the edge of the area.
He gave Uruguay the lead in the eighth with a shot from a tight angle after Diego Forlan's cross from the left. Lee Chung-yong then equalized from a header in the 68th after goalkeeper Fernando Muslera left his line to attempt to punch away a looping ball.
Crushed, the South Koreans fell to their knees at the final whistle as the Uruguayans celebrated with a group hug in pouring rain in Port Elizabeth.
The two-time world champion Uruguayans had not reached the quarterfinals at the World Cup since making the semifinals in 1970.
After Suarez scored his first goal, the South Koreans pushed forward, but Uruguay's compact defense forced them to rely on outside shots and they struggled to hit the target. Meanwhile, the South Americans sat back and waited for opportunities to stun their opponents with speedy counterattacks.
Lee Jung-soo lost the ball clumsily in the 27th — a mistake that could have been costly if Suarez had not been waved offside as he rushed toward the goal.
Uruguay was denied a penalty in the 44th when Maximiliano Pereria lobbed the ball over a defender and tried for goal.
TV replays showed his volley struck Ki Sung-yong's left elbow, but German referee Wolfgang Stark allowed play to continue.
It wasn't until after the break that South Korea found avenues into Uruguay's box. Park Chu-young sent a hard shot over the bar in the 51st and Muslera made a diving save to stop Park Ji-Sung's header seven minutes later.
Lee finally scored after Uruguay failed to clear a free kick hooked into the penalty area, and he had another chance minutes later, but his shot was too weak and aimed straight at Muslera.
Uruguay and Portugal were the only teams to go through the group stage without conceding a goal. South Korea allowed six, more than any other team that advanced to the round of 16.
South Korea reached the semifinals at the 2002 World Cup it co-hosted with Japan. But this year was the first time it advanced from the group stage on foreign soil, beating Greece 2-0, losing to Argentina 4-1 and drawing 2-2 with Nigeria.
Uruguay won its group by beating South Africa 3-0 and Mexico 1-0, and playing to a 0-0 draw with France on the opening day of the tournament.
Fernando Muslera, Diego Lugano, Diego Godin (Mauricio Victorino, 46), Jorge Fucile, Edinson Cavani, Luis Suarez (Alvaro Fernandez, 84), Diego Forlan, Alvaro Pereira (Nicolas Lodeiro, 74), Diego Perez, Maximiliano Pereira, Egidio Arevalo.
Jung Sung-ryong, Cho Yong-hyung, Park Ji-sung, Kim Jung-woo, Park Chu-young, Lee Young-pyo, Kim Jae-sung (Lee Dong-gook, 61), Lee Jung-soo, Ki Sung-yong (Yeom Ki-hun, 85), Lee Chung-yong, Cha Du-ri.
