They were helped by Belgium holding Beijing silver medallists China to a goalless draw.
Britain clearly missed their defensive powerhouse and captain Kate Walsh, who had surgery on Monday after breaking her jaw in the last minutes of the 4-0 victory over Japan but who has vowed to return later in the competition.
"The surgeon has said that her jaw has titanium plates in it and is stronger than it was previously so there is no medical risk to her playing," Britain coach Danny Kerry said.
Britain could have closed out the game earlier but allowed the Koreans to equalise twice. With 14 minutes to go and the score at 3-3 the home crowd were getting nervous before Georgie Twigg and Chloe Rogers brought Britain home.
"This team has bucketloads of character and resilience and hopefully this will stand us in good stead," said Crista Cullen.
Britain have improved in the world rankings from ninth four years ago to fourth now and are looking one of the fittest teams at the Games.
They lead the Netherlands, ranked world number one, in Group A on goal difference.
Outsiders Belgium defended superbly to hold China, who now trail the leaders by two points with three matches left to play before the semi-finals.
In Group B, two teams tipped as medal contenders lost, leaving New Zealand as the only side with two wins and four of six teams on three points.
The United States landed a surprise by beating Argentina 1-0, while Germany lost to Australia.
The United States scored late in the first half and managed to keep Argentina's captain Luciana Aymar, dubbed the Maradona of hockey, quiet for much of the match.
Not having found a way into the game in the first half, Argentina were unlucky in the second, missing several penalty corners and chances from free play.
"We tried a lot to go in from all angles, with shots, short corners. Their biggest virtue is to defend, all at the back, interrupting play and that bothers us," said Aymar, who has been voted world player of the year seven times.
"We want to score goals and make nice moves. For us it's strange to play against the United States."
The US qualified for the Olympics via a surprise victory over Argentina in last year's final of the Pan-American Games.