With the quarter-final fixture ending 1-1 in regulation time after Mumuni Shafiu scored for the Black Stars to cancel out Tiendrebeogo Ilias' early goal for the Stallions, the match was forced into penalties where the former ruled supreme.
Maxwell Konadu's fold will face Togo or Cote d'Ivoire in the semi-final of the regional championship contested by representative teams of the West African Football Union (Wafu).
The Ghana coach made three changes to his starting line-up, bringing on the Asante Kotoko trio of centre-back Habib Mohammed, midfielder Justice Blay and attacker Augustine Okrah for Issaka Mohammed, Benjamin Afutu and Appiah MacCarthy respectively.
Yannick Pognongo, who scored the solitary goal as Burkina Faso stunned the Black Stars 1-0 when the two sides met in a 2020 African Nations Championship qualifier last month, started Friday's cup match from the bench.
It took only four minutes for the first goal to be registered as Ilias timely arrived at the far post to hit a Mody Cisse cross from the right into the net to finish off a neat break.
Seven minutes to half-time, Ghana drew level through Shafiu who headed home after Joseph Esso beat two men down the left to deliver a fine cross into the box.
After both sides failed to record any more goals in a game of few clear-cut scoring chances, the match was sent straight into a penalties after 90 minutes.
Opoku Agyemang failed to convert for Ghana in the shootout but two missed efforts by Moustapha Ouedraogo and Jean Noel Lingani handed the Black Stars a 5-4 triumph.
In the day’s earlier kickoff, The Gambia advanced to the semi-final of the plate competition with a 4-2 penalty shootout victory following a 0-0 draw.
The Scorpions had the best chance of a cagey contest 12 minutes from time, but Muhammed Samba could only fire wide after being played in by substitute Momodou Bojang.
The Gambia had dropped into the plate competition after being defeated by Ghana in the first round, while Mauritania were dispatched by Burkina Faso, also following a penalty shootout loss.
The tiny West African nation will now advance to play the winner of Nigeria vs Cape Verde in the plate semi-final.