Lozano probed the South Korean defense and laid the ball off to Hernandez, who became the first Mexican global to score 50 goals for his country.
Son Heung-min managed to pull one back for Korea Republic with a stunning strike in injury-time, but it proved to be a case of too little too late for them.
Vela's penalty means more spot-kicks have now been awarded in Russian Federation than in the entire 2014 tournament in Brazil.
The goal was Hernandez's fourth at a World Cup, taking him to the same level as Luis Hernandez as Mexico's most prolific goal scorer at the premier global soccer tournament.
South Korea received four yellow cards.
Carlos Vela got the Mexicans on their way at the Rostov Arena with a 26th-minute penalty, before Javier Hernandez wrapped up the three points with a second six minutes after the hour mark. Chicharito bagged his 50th goal in worldwide 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.
The forward scored from the spot in the 26th minute after defender Jang Hyun-soo handled the ball while sliding to block an Andres Guardado cross.
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.
Miguel Layun pressed for a second goal before the break but his shot from a low cross-field drive was tipped away by Jo Hyeon-woo.
Son opens his account at the 2018 World Cup with a stunner.