Gyan ran onto a long ball from midfield, shrugged off two defenders and fired home a venomous shot that sees Ghana match the best previous performance by African teams at a World Cup.
The goal came after Ghana had been subjected to long periods of sustained American pressure with the teams having gone into extra time with the score tied at 1-1.
Ghana, who face Uruguay in the last eight on Friday at Soccer City in Johannesburg, are only the third African team to reach the quarters after Cameroon in 1990 and Senegal in 2002.
In image: Ghana's Kwadwo Asamoah, left, competes for the ball with United States' Maurice Edu, right
Coverage: FIFA World Cup