Also, Argentina held European champion Spain to 1-1 and Pakistan salvaged a 4-4 draw with New Zealand.
Yoon Sung-hoon's free kick deflected through goalkeeper James Fair's legs in the 59th minute for the only goal between South Korea and England, a day after the English edged India 3-2.
South Korea, following an opening victory over the Netherlands, became the first team in the six- nation tournament to win its first two matches.
"I think it shows our progress that we are competing - this is a team that just beat Holland, who form the number one team in the world," England coach Jason Lee said.
Argentina could be considered unlucky not to have won in front of fine 'keeping by Juan Manuel Vivaldi, as Spain was sluggish a day after beating Olympic champion Australia.
"If you don't finish your opportunities, you are going to struggle," Spain coach Maurits Hendriks said. "But what you are seeing is the level of play from the teams that aren't in the Champions Trophy has been very good. That makes it tough."
Pakistan needed Muhammed Zubair's goal five minutes from time to draw with New Zealand. A hard right cross bounced off goalkeeper Paul Woolford's chest and Zubair popped it into the goal.
New Zealand, which lost three leads during the match, believes it can secure a semi-final berth, although it has never done better than seventh at the World Cup.