Group E concludes on Wednesday, when Mexico play Sweden in Ekaterinburg and Germany play South Korea in Kazan.
Carlos Vela gave Mexico the lead with that penalty after Korean defender Jang Hyun-soo blocked Andres Guardado's cross with his arm inside the area, before the nation's all-time leading goalscorer Javier Hernandez doubled their lead in the second half with his 50th global goal.
South Korean President Moon Jae-in and his wife Kim Jung-sook looked on from the stands in the Rostov Arena as part of their three-day visit to Russian Federation, but a second consecutive 1-0 defeat leaves the 2002 semifinalists them on the brink of an early exit from the tournament.
Oddsmakers favor Mexico, most likely due to its stirring performance last Sunday in its upset win over Germany and overall World Cup pedigree - nine wins, seven draws and just two losses in its last 18 World Cup group-stage games. South Korea lost its opening match.
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Hernandez added the second in the 66th, scoring his 50th goal for Mexico.
Vela nearly turned provider two minutes later with a neat ball into Layun, but the former Watford man lashed his effort over from close range.
- Javier Hernandez is only the third player to score at three different World Cups for Mexico (2010, 2014 and 2018) after Cuauhtemoc Blanco (1998, 2002 and 2010) and Rafael Marquez (2006, 2010 and 2014).
Only a Germany win will prevent South Korea being knocked out with a group game remaining.
South Korea will face Germany at the same time in Kazan.
Shin Tae-yong's team did have one back in stoppage time when Son curled a sensational effort into the top corner, but Mexico held on in the final moments to pick up a huge win.