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Victory set up the Black Stars for a February 6 semi-final clash with the winners of Sunday's fixture between Burkina Faso and Togo in Nelspruit.
Substitute Mubarak Wakaso gave Ghana the lead in the 54th minute when he fired home a penalty after skipper Asamoah Gyan was fouled inside the box by Carlitos.
VIDEO: AFCON - Ghana vs Cape Verde - highlights
Wakaso doubled Ghana's advantage in stoppage time when he ran half the length of the pitch to slot home after Cape Verde committed every player, including goalkeeper Vozinha, upfield for a corner.
Wakaso's opening goal ignited the clash and in the 59th minute Ghana goalkeeper Fatawu Dauda denied Platini as he parried a goal-bound effort for a corner.
Cape Verde, who eliminated mighty Cameroon to reach the Nations Cup for the first time, pressed for an equaliser.
In the 82nd minute, Dauda pulled off the best save of the match when he dived full length to scramble away a goal-bound shot by substitute Djaniny.
Wakaso would have made the game safe for Ghana in the 85th minute, when he broke clear, but blasted his shot wide.
And in the 90th minute, Emmanuel Agyeman Badu was denied inside the Cape Verde box by desperate defending after Solomon Asante delivered a low cross from the right.
Man-of-the-match Dauda pulled off yet another top-class save when he scrambled to safety a last-ditch attempt by the Blue Sharks in additional time.
Ghana attacked from the onset, but it was the west African islanders who created the clearcut chances in the first half as the Black Stars found it difficult to get behind their defence.
Babanco fired from distance at the Ghana goal after just seven minutes, but missed the target as Cape Verde gradually got going with Ryan Mendes spearheading the attacking forays of the Blue Sharks.
In the 14th minute, Heldon could have given Cape Verde the lead, but blasted his shot from inside the box into the stand.
Minutes later, Dauda was forced to punch clear as Cape Verde swung a long, hopeful ball into the Ghana goal area.
Heldon, the match-winner against Angola in the final group match, again took off on a solo run in the 22nd minute, but his end-product missed the mark as the Blue Sharks began to enjoy more of the ball.
Dauda came to the rescue of the Black Stars in the 29th minute when he went down to gather a low drive delivered into the area as Cape Verde gained in confidence and surged forward.
