"Our players, I think, played a game that they will not regret", Shin added.
The opening penalty was just one of 14 spot-kicks awarded in the tournament so far - more than in the entire 2014 World Cup - with many pointing to the newly-introduced VAR as the reason for the spike in penalties.
Hector Herrera won the ball in the Korean half in the 66th. The West Ham United striker cut inside and put Mexico ahead at a time when there was growing frustration about missed opportunity. Balcazar netted for Mexico back at the 1954 World Cup.
El Tri may not have been completely clinical, but in terms of sheer grit and hard work in hot and humid temperatures - it was 33 degrees Celsius at kick-off (91 F) - against a South Korea side unafraid to break up play with tactical fouls.
Mexico had their second of the game in the 66th minute, however, when Hernandez collected a super pass from Lozano before finishing low into the bottom corner of the net to register his 50th global goal.
Mexican fans avoided using a chant, which is considered to be homophobic by Federation Internationale de Football Association, against the opposing goalkeeper.
South Korea will face Germany at the same time in Kazan.
The Taeguk Warriors of South Korea enter the Mexico contest with a seven-game World Cup winless streak. And when you take out Swansea's Ki Sung-Yun, there isn't all that much quality.
"We didn't listen to the criticism", said Javier Hernandez, one of the scorers against South Korea.
Korea's talismanic forward Son Heung-min struck a scorching consolation goal in stoppage time from outside the area.
Mexico carried their momentum from an earthquake-inducing performance on Day One, to their fixture against two-time Asian Cup champs South Korea.
Korea improved on their opening defeat by Sweden although their main tactic consisted of searching for Son with long balls behind the defensive line.