A tense match at SoFi Stadium in California appeared to be headed for extra time until Eustaquio — who plays professionally for Los Angeles FC several miles away — put a stunning volley from outside the penalty area into the bottom corner of Ronwen Williams' net.
Co-hosts Canada hung on with strong defensive play in the final minutes and advanced to face the Netherlands or Morocco in Houston on Saturday, July 4.
Coach Jesse Marsch gathered his players in a huddle after the whistle and gave a spirited speech, declaring: "You guys are Canadian heroes. Canadian heroes."
Canada played their first three matches at home in Toronto and Vancouver, but were forced to become the first World Cup host to play on the road when they lost 2-1 to Switzerland last Wednesday (Thursday AEST).
The Canadians were still the clear crowd favourites in the Los Angeles area between their travelling fans and thousands of supportive locals, including several in Eustaquio's LAFC shirt who couldn't have imagined the day they were about to enjoy.
Williams made five saves for South Africa on Sunday (Monday AEST), who managed to hang in until the final moments in search of another historic win for the team known as Bafana Bafana.
Canada had the best chances all day, but underdogs South Africa stayed firm defensively and created a handful of excellent chances themselves.
Canada finally capitalised when Alistair Johnston's long pass into the box was cleared directly into the path of Eustaquio, who struck it purely and started to celebrate almost immediately.
After finishing second in their respective groups, both South Africa and Canada were playing for their nations' first knockout victories at a World Cup when they opened the round of 32 with the only match of the day.
Canada got a boost in the 75th minute when star defender Alphonso Davies came on for his first World Cup action after missing group play with a hamstring injury.
The playmaking Bayern Munich defender missed group play after injuring his hamstring last month for the third time this year, but finally returned in the same stadium where he tore a knee ligament in March 2025 in CONCACAF Nations League play.
Davies immediately made the best pass of the day to set up Promise David for a golden chance, but he hit it wide.
South Africa reached the knockout rounds for the first time with a stunning 1-0 victory over South Korea last Wednesday.
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