Dempsey drilled the ball into the net with the goal officially timed at 29 seconds and it looked like being enough until Andre Ayew finally secured some reward for Ghana's pressure with an equaliser seven minutes from the end.
The match then appeared destined for a draw until substitute defender Brooks rose to head Graham Zusi's corner firmly into the back of the net and give the United States revenge over the team that ended their participation in the last two World Cups.
Juergen Klinsmann's side will now go onto their next match against Portugal, beaten 4-0 by Germany in the Group G opener earlier on Monday, with renewed belief that they can reach the last 16 for the fourth time in the last seven tournaments.
"We knew it was going to be a grind and it was going to go down to the wire," said German Klinsmann, who maintained his record of having won the opening match at every World Cup he has competed in as both a player and a coach.
"I was convinced we would win the game even after the equaliser. I knew two or three opportunities would come and we would have to take one. It is a good feeling to have the first three points."
The Black Stars, quarter-finalists and heroes of an entire continent in South Africa four years ago, rued their profligacy in front of goal and now face a huge task if they are to get past the group stage for the third straight tournament.
"Any team you have beaten twice will come out very strong and I believe they came out very strong and they were determined not to lose a third time," Ghana coach Kwesi Appiah said.
"We were unfortunate. I believe we played really well and had a lot of chances but couldn't take our chances."
Ghana's problems started at the back and fears many had about the fragility of their defence were confirmed when Dempsey struck before many of the fans had sat back down in their seats after the anthems.
He took a pass from Jermaine Jones at pace, cut inside central defender John Boye on his left foot, controlled the ball with his right, then went back to the left for the finish across the goalkeeper and inside the far post.
"If you score after just a minute you think there can't be anything better than that," said Klinsmann.
"But after that it maybe wasn't so good because we sat back and let them come to us. That is why I was screaming on the sideline to take control of the game."
Ghana soon showed that whatever their defensive problems, they were an altogether different proposition at the other end with right winger Christian Atsu looking particularly dangerous in all but the delivery of the final ball.
Captain Asamoah Gyan squandered a string of opportunities for an equaliser but he was not alone with Jordan Ayew, son of three times African Footballer of the Year Abedi Pele, passing up an excellent chance just before halftime with a weak shot after Atsu had beaten the offside trap.
It was left to Ayew's brother Andre to get Ghana back into the match in the 83rd minute, latching on to his skipper's neat back flick and lashing the ball into the net with the outside of his left boot from almost the same spot Dempsey had scored from.
The Americans would not be denied, however, and three minutes later Brooks proved the unlikely winner to send the red, white and blue half of the stadium into raptures.
The victory came at a cost, though, with striker Jozy Altidore looking unlikely to play any further part in the tournament after limping off with a left hamstring injury.
Dempsey finished the match with a bloodied and swollen nose after taking a whack in the face from Boye's boot, while Matt Besler was withdrawn at halftime with what looked like another hamstring problem.
It was Besler's departure that allowed Brooks to win his fifth cap, however, and Klinsmann has been at enough World Cups to know that dealing with injuries is part of the job.
"Players need to be ready, we need the whole bench," he said. "John showed his talent. What better stage to learn and grow than at a World Cup?"
(Editing by Nigel Hunt)
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