Mexico took the lead through a Carlos Vela penalty - the 14th of the tournament already - in the 26th minute after Jang Hyun-Soo's handball. With a win today against Germany, Sweden will tie Mexico for top spot in Group F, depending on goal differential.
With little else to cheer in the match, the South Korean fans were lifted by an impressive injury-time strike from Tottenham forward Son Heung-min - their first goal of this World Cup - but it was too late to push for an equalizer.
Mexico will now have its eyes on Sweden and Germany.
The South Koreans, who have lost their opening two matches, kept Ochoa under pressure throughout the match with Son often out-sprinting the Mexican defense.
Given the form the team has shown so far this year in Russian Federation, however, fans will no doubt be disappointed if Mexico doesn't at least make it through to an elusive fifth match for the first time since the World Cup was last held in Mexico in 1986.
South Korea coach Shin Tae-yong stuck to his attacking line up but shook up his midfield with Ju Se-jong and Moon Seon-min coming in for Park Joo-ho and Koo Ja-cheol, as the 2002 semifinalists seek to recover from their opening 1-0 loss to Sweden.
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The crowd numbered in the tens of thousands, with organizers estimating 100,000 people took part. The UK is yet to define exactly what it wants in its future relationship with the EU.
Ki Sung-yueng threatened in the 56th minute with a ferocious drive that Guillermo Ochoa could only gather at the second attempt, before Cho did well two minutes later to keep out a curler by Guardado at full stretch.
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.
Mexico has reached the round of 16 in six straight World Cups, but the team has not advanced to the quarterfinals since playing at home in 1986.
The Taeguk Warriors of South Korea enter the Mexico contest with a seven-game World Cup winless streak.
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.