Goals from Yaw Yeboah, Dauda Mohammed and Evans Mensah wrapped up a 5-2 aggregate at the new Cape Coast Sports Stadium, the first leg having ended 2-1 in favour of the Ethiopians in Addis Ababa.
“I knew what I had taught them going into the game, so we were just going onto the pitch to play our game," Dramani said.
“We had two early goals which were so important, and then the other goals came in the second half.
"I want to congratulate the players and I know we have to work harder because there is more work to be done.”
Manchester City youngster Yeboah, who opened the scoring on six minutes, added: “We worked hard and listened to our coaches.
"We have been training very hard because we knew this game was very important. They came to get a draw but we did everything possible to win.
"I will say hard work made us score the four goals."
Ghana have now set a date with Senegal in the third and final round.
Seven teams are expected to join host Zambia for the final championship which runs between February 26 and March 12, 2017.