It marks a major triumph for a team that failed to get out of the group stage in 2022 and now has won twice on home soil in front of jubilant crowds.
Luis Romo scored in the 50th minute after South Korea goalkeeper Kim Seung-gyu collided with defender Lee Gi-hyuk and dropped the ball inside the area. Romo easily found the open net after picking up the loose ball.
The South Koreans nearly equalized in the 87th minute when Mexico goalkeeper Raúl Rangel stopped a header from close range by Cho Gue-sung, then made an even better save on the rebound, extending his right arm to keep the ball from crossing the line.
Mexico has six points from two Group A matches, three more than South Korea and five more than the Czech Republic and South Africa, who drew 1-1 earlier Thursday in Atlanta.
The top two teams from each group move on to the knockout stage, along with the best eight third-place teams. A round of 32 is being played for the first time at the World Cup after the tournament was expanded to 48 teams.