Ghana upped the tempo and equalised when Andre Ayew latched on to Asamoah Gyan's backheel to fire in a leveller.
"One of the moments of the World Cup for me was the reaction of John Brooks for the winning goal. That was priceless. That is what the World Cup is all about.
"The way Ghana conceded the corner kick that led to the USA winner was poor. In these games, you can't do that. They are in trouble. They still have to play the big dogs in the group."
USA looked tired but Brooks nodded in from six yards following a corner to give his side a winning start.
Centre-back Brooks reacted with a mixture of disbelief and delight, the Germany-born 21-year-old marking his Fifa World Cup debut and fifth cap in style.
The victory also gave USA a measure of revenge for being knocked out of the past two World Cups by Ghana.
They had looked to be tiring, and a winner seemed unlikely as Ghana dominated the second half and pulled themselves level.
But their tenacity was rewarded to leave the Black Stars precariously placed, with both teams having to face Germany and Portugal.
Ghana exited the World Cup four years ago in tears, when Gyan's missed penalty in the last minute of extra-time against Uruguay denied them the chance to become the first African side to reach the semi-finals.