Luis Romo scored the only goal early in the second half, capitalising after South Korea goalkeeper Kim Seung-gyu fumbled a routine situation and gifted the midfielder a simple finish.
It was a cagey contest for long periods, with Mexico dominating the early stages before South Korea grew into the game and finished with a marginally superior expected-goals tally of 0.69 to 0.48.
South Korea's best chance of an equaliser came deep into stoppage time, when Lee Kang-in's curling corner picked out Lee Han-beom at the far post, but the header drifted just wide of goal.
The result means Mexico have already secured qualification for the last 32, the first side to do so at this World Cup, and they will finish the group stage against Czechia in Mexico City.
South Korea remain in second place for now, with their fate to be decided by a final group game against South Africa in Monterrey.
Mexico head coach Javier Aguirre reflected on a difficult, low-scoring contest.
"It was difficult. We know them very well. They put us under a lot of pressure. They didn't give us any space, and neither did we," Aguirre said. "In the end, it felt like one mistake was always going to make the difference one way or the other. It wasn't a great match, and the opposition didn't allow us to do much."
South Korea coach Hong Myung-bo took some encouragement from his side's response despite the result.
"We played just as we planned. The way we conceded was disappointing," Hong said. "We will give everything in the last game of the group stage. I asked the players to remain calm and play our game. It wasn't bad. We kept our composure throughout the game. We will focus on preparing for the next game now."
The win extends Mexico's perfect start to the tournament as co-hosts, having opened with a 2-0 victory over South Africa in a match that produced three red cards. South Korea must now beat South Africa in their final fixture to guarantee progress, with their path to the knockout stage no longer entirely in their own hands depending on how results elsewhere fall.
For Hong's side, the manner of the defeat will sting more than the scoreline. South Korea matched Mexico for long periods and arguably created the better chances, but a single lapse from their goalkeeper proved decisive in a tournament where fine margins are increasingly defining outcomes.