In a fractious game in Blida a solitary Madjid Bougherra goal decided the tie, with the visitors failing to greatly trouble the hosts as they had chased their debut appearance at the global finals.
Instead Algeria make it successive appearances at the competition.
With Burkina Faso having a lead to protect as the game began, but only a goal needed for Algeria to put them through, the first-half was a cagey affair.
There was little between either side as Burkina Faso dug deep and neither side were able to get any real rhythm in their play.
The best chance of the opening period fell to the head of Islam Slimani. Faouzi Ghoulam worked space for himself down the left-wing, before whipping a ball on to the head of the Sporting forward, but his stooping effort flashed wide of Daouda Diakite’s goal.
There half was more characterised by the tough tackles that were flying in for either side, with Algeria captain Bougherra coming out of the opening 45 minutes with a yellow card to his name.
After the break though the Lekhwiya defender broke the deadlock in the scrappiest of fashions just four minutes in.
Ghoulam’s free-kick got the better of what had been a largely solid Burkina Faso rearguard, with a scramble ensuing. Bougherra’s initial effort was initially saved, but as Bakary Kone tried to clear, it ricocheted in off the grateful Algerian.
With the North Africans now on their way to Brazil, with the score as it was, the visitors began to open up in search of a goal that would swing the pendulum back in their direction.
Even so, the CAF Africa Cup of Nations semi-finalists were struggling to test Mohamed Lamine Zemmamouche.
It was in fact Slimani who went close again next, with his swivelling effort going just wide following Sofiane Feghouli’s flicked header.
Substitute Aristade Bance caused trouble when he arrived to bolster the Burkina Faso attack, but they almost went through in dramatic style as a late corner is deflected onto his own bar by an Algerian player, but the hosts hung on for the win.