Trailing 6-1 from the first leg last month, Egypt took a 2-0 lead on goals by Amr Zaki in the 25th minute and Gedo in the 84th. Kevin Prince-Boateng scored for Ghana on a counterattack in the 89th.
Egypt has not qualified for the World Cup since 1990, and Bradley said he would leave if his team didn't qualify. Ghana also eliminated Bradley's U.S. team in the second round of the 2010 tournament.
A quarterfinalist three years ago, Ghana joined Cameroon, Ivory Coast and Nigeria as Africa's representatives at June's World Cup in Brazil. Burkina Faso had a chance to make it a sweep for west Africa when it played Algeria later Tuesday.
About 30,000 were at Air Defence Stadium for the first international match in the Egyptian capital since 2011. It was among the first games in Egypt with a crowd of that size since the Port Said riot last year, when more than 70 people were killed.
It was played on the second anniversary of some of the worst clashes between protesters and police near Cairo's Tahrir Square, on the other side of the city. While there were some scuffles near the square Tuesday, there appeared to be no trouble at the game.
About 200 Ghanaian fans in the stadium sang and waved flags to celebrate their country's qualification.