Vahid Halilhodzic's side headed into the game knowing that they had done much of the hard work in their bid to clinch a second successive appearance at the World Cup finals, following two crucial goals in a 3-2 defeat in Ouagadougou in the first leg.
Burkina Faso, by contrast, were looking to qualify for the tournament for the first time.
But Paul Put's men were denied the chance to etch their place in history, as Algeria deservedly progressed by the narrowest of margins in a game that they dominated throughout.
The hosts netted the crucial goal four minutes after half-time, captain Madjid Bougherra credited with the winner, although the final touch looked to have come off Burkina Faso defender Bakary Kone.
Put's men created very few opportunities in a disappointing display, and they were left to rue what might have been as Algeria held on to spark scenes of celebration in Blida.
Mohamed Zemmamouche replaced Rais M'Bolhi in the Algeria goal as one of five changes from the first leg made by Halilhodzic.
Burkina Faso, meanwhile, made four alterations to their starting XI from the encounter in Ouagadougou, with Toulouse defender Steeve Yago among those coming into the side.
The visitors made the better start to proceedings and had the ball in the net inside two minutes, only for Charles Kabore's strike to be ruled out for offside.
Algeria slowly began to dictate play, though, and they should have taken the lead after 22 minutes when Islam Slimani headed wide from Faouzi Ghoulam's left-wing cross.
Chances proved rare as Burkina Faso defended stoutly throughout the first half, although the hosts did test goalkeeper Daouda Diakite with a long-range effort from Yacine Brahimi on the stroke of half-time.
But just four minutes into the second half, Algeria did take the lead to level the tie, Bougherra wheeling away in celebration after his effort appeared to have been diverted into the net by Kone following a goalmouth scramble from Ghoulam's right-wing free-kick.
Halilhodzic's men continued to pile on the pressure, and they went close to finding a second as Slimani volleyed wide just before the hour mark.
Algeria restricted Burkina Faso's possession in the closing stages, with substitute Alain Traore's tame volley summing up the visitors' inept attacking performance.
