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 quarterfinals 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 said. 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 quarterfinal in Cabinda on Sunday.
Nigeria eye quarters
Nigeria, who will reach the African Nations Cup quarter-finals with at least a draw against Mozambique in their last Group C game in Lubango, believe they are hitting their stride.
The Super Eagles are in second place in the group on three points, three behind Egypt, who will face Benin in Benguela.
"We lacked confidence against Egypt in our first game (a 3-1 defeat) but were much more relaxed when beating Benin (1-0) and I believe we will deliver an even better performance against Mozambique," said Nigeria striker Peter Odemwingie.
"Mozambique are a team we know, having played them twice last year and I am confident we will give our best display so far in Angola, win the match and advance to the last eight." added Odemwingie.
The last meeting between Nigeria and Mozambique ended in a goalless stalemate in a World Cup qualifier in Maputo last March.
Benin and Mozambique, both on one point, have never made it past the first round at the African Nations Cup.