The forward hit the winner in first-half extra time.
Gyan scored the winner as he fended off his opponent and volleyed home in the 93rd minute.
"We are the third African nation that has qualified for the quarter-finals," Gyan told Associated Press after the match. "We have made everybody proud, not only Ghana, but the whole of Africa."
Ghana was the only African nation to qualify for the knockout stage of the tournament, after Nigeria, Cameroon and hosts South Africa all failed to progress past the group stage.
The Black Stars will now take on Uruguay in the quarter-finals.
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