Mexico claims spot in World Cup last-32 to set up last day scramble for glory
Luis Romo scored the only goal of the game at the Estadio Akron in Guadalajara.
The victory gave Mexico six points and automatic qualification for the last 32 of the 48-team tournament.
Following South Africa's 1-1 draw with Czech Republic in Atlanta, South Africa could advance as runners-up if they were to beat South Korea on 24 June in Monterrey.
However, a draw would be enough for the Koreans to move into the last 32.
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The Czech Republic also have a chance to finish second in Group A.
But they need to beat Mexico handsomely and hope that South Korea lose to the South African who gave themselves a lifeline with the stalemate.
Michal Sadílek scored the Czech Republic's opener against South Africa in the sixth minute and his side held onto the advantage until seven minutes from time.
Thapelo Maseko headed towards the goal from the right wing and crossed the ball which hit Pavel Sulc's arm in the penalty area.
Mokoena salvages point for South Africa
Referee Tori Penso pointed to the penalty spot and Teboho Mokoena converted the kick.
South Africa boss Hugo Broos hailed the resilience of his players who lost 2-0 to Mexico in the opening game of the 2026 tournament. They also had two men sent off.
"We played a very good game," said Broos. "And you need a little bit of luck also sometimes in the game. That was not the case for us," added the 74-year-old Belgian.
"When we shot at goal, there was always a leg or a body which could block the shot.
"But if we play against South Korea like we did against the Czech Republic, then I think we'll have a chance to go into the second round."
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South Africa have never tasted such lofty heights in their three appearances at the World Cup.
In 2010, they became the first hosts in the then 80-year history of the competition to be eliminated after the first phase.
Qatar suffered the same fate in 2022.
"The guys were very determinated," added Broos. "And they showed a lot of perseverance in the game. They wanted to win the game and that makes me very happy.
"I'm very proud of the performance of my team. And I'm very proud of my team."
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South Africa conclude their Group A campaign against South Korea in Monterrey on 24 June while Mexico take on Czech Republic at the Estadio Azteca in Mexico City.
In other games on Thursday night, in Group B, Jonathan David hit a hat trick as Canada walloped nine-man Qatar 6-0 to take charge of the pool.
It was the country's first victory at a World Cup. The success came on their third visit to the most prestigious international team tournament and sets them up for a winner-takes-all clash with second-placed Switzerland who swept past Bosnia and Herzegovina 4-1 on Thursday.
In the other final pool game in Group B, Qatar take on Bosnia and Herzegovina. The winner of that match could feature in the second phase.