Dede Ayew, whose father Abedi Pele was the pivotal figure when Ghana last won the trophy in 1982, headed the only goal on the half-hour to help his side finish second behind Group B winners Ivory Coast.
Burkina Faso, who needed a draw to qualify, were eliminated. They had 10 men for the last 24 minutes after Mamadou Tall was sent off.
Ghana now play hosts Angola in the quarter-finals in Luanda on Sunday.
After the unmarked Ayew struck, Ghana squandered several chances to add to their goal tally.
Burkina Faso, who finished bottom of a group reduced to three teams after Togo's withdrawal following the deadly gun attack on their bus, tried to force an equaliser but only winger Jonathan Pitroipa was able to penetrate Ghana's defence and create half-chances.
Tall was dismissed after he collected a second yellow card.
"It was a very important match for our young team and it was both physically and psychologically tough," Ghana coach Milovan Rajevac told reporters.
Rajevac lost several players before the start of the tournament and has since seen captain Michael Essien injure his knee in training.
Essien watched from the stands after deciding to stay and support his team mates rather than return immediately to his club side Chelsea.
Ivory Coast will meet Algeria in their quarter-final in Cabinda on Sunday.
