Ghana led 2-0 at halftime but allowed South Korea back into the match as Cho Gue-sung netted twice in three minutes to level at 2-2, only for Kudus to earn Ghana victory in a heart-stopping contest at the Education City Stadium.
Ghana, who lost their opening group game 3-2 to Portugal, moved on to three points in the group, with the Koreans on one after two games.
Ghana saw off early Korean pressure to go 2-0 ahead as centre back Mohammed Salisu's scrambled goal in the 24th minute was followed 10 minutes later by a glancing header from Kudus.
Both were at the end of superb deliveries from Jordan Ayew, restored to the starting line-up as Ghana fielded a much more offensive line-up than against Portugal.
Ayew's perfectly-weighted free kick caused confusion in the Korean defence and Salisu turned sharply to snap up the opener as the ball bobbled around. There was a lengthy wait for VAR to check the validity of the goal before English referee Anthony Taylor confirmed it.
Ayew again provided a dangerous delivery, with the perfect height and pace for Kudus to apply a glancing touch with his head and double the Black Stars' lead.
Early second-half changes breathed new life into Korea's attack and Lee kang-in had barely been on the field for a minute when he helped dispossess Tariq Lamptey and fired in a cross that Cho attacked with pace, leaving Salisu watching as he got in front of his marker to power home a 58th-minute header.
The 24-year-old centre forward was even more determined as a chip from Kim Jin-su caught the out-of-position goalkeeper Lawrence Ati-Zigi scrambling and Cho climbed above the centre backs to score a dramatic equaliser, right in front of a bank of screaming Korean supporters.
Ghana, who would have been eliminated from the World Cup had they lost, looked vulnerable but were handed a surprise lifeline by sloppy Korean defending in the 68th minute.
Gideon Mensah was allowed to get a cross in from the left, which Inaki Williams missed but the ball fell for Kudus to tuck home with his left foot - all while the Korean backline failed to attack the ball.
"We showed the same mentality today as we did against Portugal. The guys were awesome today and we won the three points," said Kudus.
The fortuitous turn for the Black Stars was followed by Ati-Zigi making a fine save from Lee's free kick and watching as the ball was cleared off the line from Kim Jin-su's scuffed shot.
Ghana continued to live dangerously through 10 minutes of added time as Korea besieged their goal, but were unable to equalise again.
The referee was besieged at the final whistle as he ended the game with South Korea about to take a corner and their coach Paulo Bento earned a red card for his protestations.
"Big congratulations to the team to come back and score a third and then the passionate defending, South Korea put on a lot of pressure ... in the end we were a bit lucky," Ghana coach Otto Addo said.