Bance had a first-half spot kick saved but bravely stepped up to score in the 86th minute and hand the Burkinabe a slender lead for the return leg in Blida on Nov. 19 which will decide who gets a place at next year's finals in Brazil.
The penalty was awarded for handball on the advice of the linesman but television replays showed defender Essaid Belkalem clearly had his arms tucked behind his back and blocked the shot with his chest.
Algeria had twice come from behind to level the score in the second half through Sofiane Feghouli and Carl Medjani and will be bitter about not coming away with a draw as they squandered several other good chances after the break.
Feghouli tucked away a right-foot finish inside the area in the 50th minute after Jonathan Pitroipa had given Burkina Faso the lead with the last touch of the first half.
Pitroipa sneaked between two static defenders to head home a cross from left winger Jean-Noel Lingani two minutes after Algeria's Rais Mbolhi had saved Bance's first penalty.
Burkina Faso's Djakaridja Kone put the hosts back in front in the 65th when he intercepted the ball in midfield before striding out wide on the left and tucking the ball across Mbolhi into the far corner of the net.
But their advantage was again short-lived as Medjani rose above the defence to head home a corner four minutes later.
Saturday's match was the first of five African World Cup playoff first legs over the next four days.
Later on Saturday, the Ivory Coast host Senegal while on Sunday Ethiopia host Nigeria and Tunisia welcome Cameroon before Ghana play Egypt in Kumasi on Tuesday.