Bounedjah netted from a corner five minutes into stoppage time at the end to ensure his side a share of the spoils and deny Burkina Faso early progress to the next stage.
Bounedjah scored both Algeria's goals while Mohamed Konate and Bertrand Traore netted for the Burkinabe, who remain top of the group on four points compared with two for Algeria.
Burkina Faso came desperately close to a smash-and-grab victory that would have seen them through to the last-16 with a guaranteed top two finish. They were under pressure for much of the match but had two VAR decisions go in their favour.
Algeria failed to take their first-half opportunities, with Youcef Belaili wasteful as Burkinabe goalkeeper Herve Koffi looked nervous and twice cleared balls straight to the Algerians and was fortunate they did could not finish.
Almost inevitably, Burkina scored with a rare attack in stoppage time at the end of the first half as a long ball from Abdoul Fessal Tapsoba went over the Algerian defence and was headed homed by a leaping Konate. The whistle was immediately blown for offside, but a VAR check went in Konate's favour.
Algeria equalised in the 51st minute when a free kick was partially blocked and fell for Nabil Bentaleb, whose shot was parried by the goalkeeper down to the feet of Bounedjah, who recovered quickly to sweep the ball in from close range.
Burkina Faso's lead was restored from the penalty spot in 71st minute by substitute Traore, who had also netted a stoppage time penalty for a 1-0 win over Mauritania in their first Group D game.
It was another VAR check that went Burkina's way, revealing a wild tackle by Rayan Ait Nouri on Issa Kabore. The referee had not given the kick but replays showed Ait Nouri had missed the ball as he attempted to clear and caught Kabore on the shin.
A corner late in the game delivered relief for Algeria as Bounedjah leaped up to head home from the set piece.
Algeria were Cup of Nations winners in 2019 but made a horror defence of their title in Cameroon two years ago, where they went home with a single point and in last place in their group and faced a similar scenario again after drawing their opening game in the Ivory Coast against Angola