The Algerians struck early in the second half through Madjid Bougherra - who should have seen red for a horrific lunge minutes earlier - as the ball deflecting in after Bakary Kone's attempted clearance struck him on the head, as the tie finished 3-3 on aggregate.
Burkina Faso, bidding to qualify for the first time, went agonisingly close late on when they hit the post.
Algeria join Cameroon, Ivory Coast, Ghana and Nigeria as Africa's World Cup representatives.
The Mustapha Tchaker stadium was full six hours before kickoff and the sense of expectation from the fans translated to the players as the home side made a cautious start.
Lady Luck smiled on the hosts three minutes into the second period when Ghoulam's free kick was not cleared by the Burkinabe defence and the ball fell kindly for Bougherra three metres from goal.
His initial effort was saved by Daouda Diakite and as Kone tried to hook the ball away he succeeded only in blasting it against the forehead of Bougherra and it rebounded into the net.
Slimani then flashed a shot on the turn just wide as Algeria, buoyed by the jubilation in the stands, pushed forward in search of a second goal.
The remainder of the game was largely played in the Burkina Faso half but there was late drama when a corner from the visitors was deflected on to the Algeria post and bounced away to safety.
- Sports & Recreation
- Madjid Bougherra
- Burkina Faso
- Bakary Kone