South Korean forward Son Heung-min scored a scintillating consolation goal in injury time but it was too little too late for the East Asian nation, which now can not qualify for the first knockout stage of the tournament.
Mexico will clinch their place in the next round if Germany fail to beat Sweden in the other Group F game later on Saturday. "We are going to enjoy it", Hernandez said.
The Mexican Football Federation was slapped with a fine of 10,000 Swiss Francs (US $10,120) after fans shouted the homophobic taunt during the team's 1-0 victory over Germany last Sunday.
But, six minutes before the half-hour mark, El Tricolor were handed the flawless opportunity to open the scoring when opposition defender Jang Hyun-soo needlessly went to the ground in an attempt to block an Andres Guardado cross - only to handle the ball inside the area.
A free kick by South Korea goalkeeper Jo Hyeon-Woo in the first half was greeted with different chants by the Mexico fans, who made up the vast majority of 43,000 supporters in Rostov Arena. "Our talent and hard work have paid off".
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Roman protested that she hadn't seen any warning signs, and didn't initially anticipate how serious the matter would become. She had no documentation on her person.
Conditioned by the dry heat that has been a feature of every game at the Rostov Arena, Mexico did not quite have the same thrust and energy as against Germany although they still controlled most of the play and looked far more risky.
"I think our players had a game they will not regret", South Korea coach Shin Tae-yong. A volley from Vela skimmed the bar minutes later. "All of the players became one, and they did their utmost until the very last minute". Mexico relied on keeping possession and long-range shots, with a powerful attempt from Andreas Guardado forcing an athletic save from Jo in the second half.
Despite their overall boisterousness, Mexico's fans didn't use their usual controversial "eh, puto!" chant during goal kicks taken by the South Korean goalkeeper.
After the match - as is usual - passionate soccer fans spilled into the streets all over the country to celebrate the victory, including Mexico City where the capital's annual gay pride march is also being held today.
The record for penalties at the World Cup stands at 18 in 2002.
South Korea will face Germany at the same time in Kazan.