Javier Hernández scored the second after 66 minutes, finishing off a rapid break.
ROSTOV-ON-DON, Russia-There were plenty of teams to be afraid of at the start of this World Cup.
With the second goal Mexico tried to shut up shop by sending on the experienced Rafael Marquez for Guardado.
Son Heung-min's stunning long-range effort in stoppage time gave South Korea a glimmer of hope but it was too little too late.
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South Dakota had challenged the physical-presence rule, saying it was losing roughly $48 to $58 million in revenue each year. They could lower "the sales tax rate or make more significant investments in infrastructure which creates jobs", he says.
The South Koreans managed successive counterattacks by targeting striker Son Heung-min but caused few problems for Mexican goalkeeper Guillermo Ochoa. Chicharito bagged his 50th goal in global play on a counter-attack led by breakout star Hirving Lozano - but Mexico was only able to score because referee Milorad Mazic allowed play to continue after Hector Herrera fouled Ki Sung-yueng to gain possession.
South Korea had been pegged back from the first whistle.
However, Jang Hyun-soo's handball gave referee Milorad Mazic little choice but to point to the spot - the number of penalties awarded in Russian Federation consequently surpassing the 13 across the entirety of the previous World Cup - and Vela, who announced the death of his grandfather following the win over Germany, converted.
If Germany lose to Sweden in the evening's Group F game at the Fisht Stadium in Sochi then Joachim Löw's side will be out.
The 20-year-old Alvarez, one of Mexico's hottest prospects, replaces Hugo Ayala.
Mexico will finish their group-stage campaign against Sweden on June 27, while South Korea face the world champions Germany on the same day.