They clinched the victory after Emmanuel Agyemang Badu missed Ghana's third penalty after a weak shimmy and shot.
The Burkinabe will be without the influential Jonathan Pitroipa in Sunday's final against Nigeria after he was sent off three minutes from the end of extra time for a second yellow card.
Ghana went ahead after 13 minutes when Mubarak Wakaso scored his third penalty of the tournament after Burkina Faso defender Paul Koulibaly was harshly judged to have fouled Christian Atsu.
Burkina Faso equalised after exactly an hour when Aristides Bance fired home low and hard from the edge of the penalty area.
The shootout started badly for Ghana after Isaac Vorsah sent his spot kick wide and Burkina Faso kept the upper hand as Bakary Kone, Henri Traore and Bance all scored.
Ghana were fortunate to be awarded their 13th-minute penalty by Tunisian referee Slim Jdidi when there appeared little or no contact between Koulibaly and Atsu even though Atsu went to ground after they challenged for a high ball.
Wakaso made no mistake with his fourth goal of the tournament and the third from the spot, placing the ball well out of the reach of the diving Daouda Diakite.
The Burkinabe responded well to the early setback with Jonathan Pitroipa a constant menace to the Black Stars' defence.
They crafted a superb opening after 31 minutes, too, when a pinpoint long ball from midfield was initially brilliantly controlled by Prejuce Nakoulma who then wasted his good work by blasting the ball high over the Ghana bar.
Burkina Faso came far closer to an equaliser after 53 minutes when the distinctive Bance, with his dyed blond hair, powered a header goalwards from a corner, only to see goalkeeper Fatau Dauda clutch it from under the bar on the line.
Five minutes later, with Burkina Faso pushing up in search of an equaliser, Atsu set up Asamoah Gyan for a chance to double Ghana's lead but his shot rebounded away off a post, and that proved costly because, minutes later, Burkina Faso equalised.
Bance was not to be denied a second time and he made no mistake with his next opportunity after an hour when he fired home from close range to equalise.
Both teams went close to a winner which never came in the last half-hour, forcing the game into extra time, when Bance missed a real opportunity to win the game, but blasted high over the bar in the 104th minute.
Burkina Faso did have the ball in the net just before the extra-time break, but the goal was disallowed because of a foot up on the goalkeeper from Nakoulma.
Bance, who had eight shots in all, saw a second effort saved off the line after 112 minutes by defender Harrison Afful.
