Eltaib Mudather's brace saw Sudan complete the turnaround they required in African Nations Cup group B and reach the knockout stages.
The striker scored once in each half and Angola's 2-0 defeat to the Ivory Coast ensured Sudan progressed in second place behind the Elephants.
Sudan went into the game needing to win and hope Angola lost - with at least a three-goal swing - if they were to qualify.
But they lost Najem Abdullah to a serious injury in the early stages after a collision with Akram El Hadi as the goalkeeper dived to save at the feet of dangerman Jonathan Pitroipa. Omar Moseab took Najem's place.
Moumouni Dagano's header for Burkina Faso clipped the post and the Stallions captain then curled a free-kick over before again heading wide.
Mohamed Tahir had Sudan's first serious effort with a free-kick after he was fouled by Djakaridja Kone, but his shot was deflected wide.
But the 33rd minute brought double delight for the Falcons of Jediane as both they and the Ivory Coast took the lead almost simultaneously.
As a sliding Emmanuel Eboue netted in Malabo, Mudather ran onto Ahmed Bashir's fine first-time pass and evaded a dreadful two-footed challenge from Hamadou Tall before finishing calmly to give his side the lead at the Estadio de Bata.
Burkina Faso could have been reduced to 10 men when Mady Panandetiguiri lunged in two-footed on Alaadine Yousif, but the defender escaped with a booking.
Pitroipa missed a chance to equalise before half-time when he was played in by Dagano but finished weakly.
Burkina Faso substitute Issiaka Ouedraogo missed a glorious chance to equalise in the 56th minute, just two minutes after his introduction, poking wide when through on Akram.
Dagano then headed the lively Ouedraogo's cross just wide from outside the penalty area and Sudan's Ahmed Khalifa was off target with an overly ambitious effort.
Bony Wilfried's goal for the Ivorians swung the qualification race towards Sudan and back in Bata, an offside flag ruled out a Dagano header which would have changed the situation once more.
Muhannad Tahir tried to make the result safe with a snap-shot which was tipped over by keeper Daouda Diakite.
With 10 minutes remaining, though, Mudather did make it 2-0 via route one, racing onto Akram's long clearance and rounding Diakite to tap home.
Ouedraogo grabbed a consolation goal, heading home Pitroipa's long ball deep into stoppage time, but it was not enough to deny Sudan their qualification.