A goal from stand-in captain Jan Vertonghen with 12 minutes remaining handed the Belgians their third straight triumph and condemned the Far East team to an early exit from the competition.
The Red Devils also had Steven Defour sent off for a reckless challenge late in the first half, but despite playing with 10 men, Wilmots felt his side dominated at the Arena de Sao Paulo.
"We knew all games were going to be tough, but even with 10 men we still pushed forward and won the game. These guys are really hungry," he said.
"I particularly liked the mentality that we've had when we were down to 10 men.
"Despite the initial shock after Steven's sending off, the players stayed in the game."
Wilmots praised the work done by second-half substitutes Nacer Chadli and Divock Origi.
"Again the difference was in the last quarter of an hour," he said. "Divock and Nacer brought their speed and penetration.
"The South Koreans needed to come out and had to take risks.
"We took advantage."
Wilmots believes his side has so far delivered on the expectations the team had going into the tournament.
Belgium have scored four goals and given away one in the group stages heading into Tuesday's round of 16 game against the United States.
"There was a lot of pressure for my players ahead of this tournament in Brazil," Wilmots said. "I think with this nine out of nine, we have responded well."
Stand-in captain Vertonghen was delighted to score at a World Cup.
"This is a special moment for me," said the Tottenham defender, who was voted Man of the Match. "It's simply magical.
"I scored as though I was a true striker.
"I took advantage of Divock's rebound to beat the Korean goalkeeeper.
"We have done something historic by taking nine out of nine from the group stages.
"We worked very hard and deserved the win."