Seoul and Washington say the rocket launch is a disguised long-range missile test, and have deployed missile-interceptor ships ahead of the launch, scheduled to take place from April 4-8.
Putting aside the political tensions, South Korean fans stood for the anthem of rival North Korea before the kick-off at Seoul's World Cup Stadium and politely cheered the opposing side.
The two Koreas are still technically at war.
South Korea's goal came in the 87th minute when a left-footed free kick from substitute Kim Chi-woo slid off the hand of North Korean goalkeeper Ri Myong-guk into the net.
With the win South Korea leap-frogged North Korea to take top spot in Asia's Group 2. North Korea are trying for their first World Cup finals since their surprise run to the final eight in 1966.
South Korea dominated play but failed to break through despite numerous chances including two close-range shots from Lee Keun-ho and several long-range shots by forward Park Chu-young.
North Korea's coach Kim Jung-hun was bitter after the match, saying the World Cup qualifier had been played in unfair conditions.
Kim said he had requested to move the game to a third country at a future date because three of his players had suffered food poisoning.
South Korea's soccer federation said the players had been seen by physicians before the kickoff and did not appear to display serious symptoms. They refused to go through further examinations, the federation said.
Kim also criticised the referee for denying his side a goal. South Korean goalkeeper Lee Woon-jae placed his hand on the ball after a header by the North's Jong Tae-se appeared in replays to cross the goal line.
"As a coach, I have a lot to say about the referees," Kim told a news conference. "It obviously crossed the (goal) line but they ignored it. Referees should be fair," he said before storming out of the room without answering questions.
SPORT AND POLITICS
Before the game, both coaches said they would like to see the two Koreas advance for the first time together to the World Cup finals, which will be played in South Africa next year.
"I expected it would be a one-goal margin," said South Korean coach Huh Jung-moo after the match. "I felt the North's team were getting better and better and they will be an even better team."
Matches between the two have typically been low-scoring, dour affairs.
North Korea also crossed the line between sport and politics early last year when it refused to play the South's anthem or raise its flag in an earlier round of qualifying in Pyongyang.
That decision forced world soccer's governing body FIFA to change the venue to Shanghai, where the two teams played out a draw.
Tensions on the peninsula have been raised by North Korea's plans to launch the rocket and threats from Pyongyang that any interference from South Korea or its allies would be seen as a declaration of war.
South Korea, the United States and Japan, have said the launch i
