Riyad Mahrez scored the only goal of the game at the Moulay Hassan Stadium with a 23rd minute penalty to hand his side the all-important three points.
The result sees Algeria move onto six points following their opening day Group E win over Sudan while Burkina Faso, on three points, will need a result in their final game with Sudan to seal their place in the knockout rounds.
Algeria had started the stronger of the two sides but there were calls for an early penalty for Burkina Faso as Bertrand Traore went down in the box under a challenge from Ismael Bennacer, but after a quick review, play was waved on by the referee.
A heavy tackle by Gustavo Sangare on midfielder Jaouen Hadjam briefly halted play again - for a VAR review on a potential red card, that was also dismissed - with the player picking up a yellow card instead.
The game continued momentarily, but the tackle on Hadjam had done its damage, and the player was forced to leave the field with a rolled ankle, and replaced by forward Baghdad Bounedjah.
Algeria's Ibrahim Maza scored in the win over Sudan and got his first shot away in this tie on 15 minutes that was calmly collected by Burkina Faso goalkeeper Kouakou Koffi.
Aissa Mandi was then well placed to head clear a dangerous cross at the other end from Dango Ouattara in his first real attacking involvement in the tie.
But it was Algeria who were to break the deadlock, after just 23 minutes as Mahrez slotted his spot kick into the bottom left corner of Koffi's goal after Ismahila Ouedraogo had upended upending Rayan Ait Nouri inside the box.
Burkina Faso responded with Bertrand Traore dancing his way into the box and drawing an excellent covering tackle from Ramy Bensebaini.
Pierre Kabore was then just wide with a volley as a 27th minute Traore free-kick dropped to him on the edge of the area, and the same player then wasted a great chance to get a shot away just a minute later after being teed up off the head of Ouattara.
Amoura fired over from close range with 35 minutes played as Algeria looked to extend their advantage against an ever-improving Burkina Faso side.
Kabore was next with a goal bound shot that was blocked before Ouedraogo was dispossessed as he turned well on the edge of the Algeria box.
A 42nd minute corner from Traore caused all sorts of problems as a header from Ouattara came back off a post before being scrambled out of play.
And although Algeria started the tie the stronger of the two teams, they would have been happy to hear the half-time whistle.
They were, however, close to extending their lead in stoppage time after a swift counter and shot from Amoura that Koffi did well to get his body behind.
Georgi Minoungou, who scored the equaliser in the win over Equatorial Guinea on matchday one, replaced Mohamed Zougrana at the interval.
The start of the second half was a more even affair with both teams probing for space and a potential opening, but chances were few and far between.
A 55th minute Gustavo Sangare free-kick from the left of the Algeria goal was easily headed clear by Ramy Bensebaini before a 60th minute set-piece was called up for offsides as Mandi looked to feed recent substitute Zineddine Belaid, who came on for Mahrez.
Ismahila Ouedraogo had options to his left and right when the ball dropped to him just past the hour mark, but he opted to shoot and give Algeria stopper Luca Zidane an easy save.
Lassina Traore came on for Kabore after 66 minutes as Burkina Faso looked for a way back into the game, while almost immediately Amoura blasted wide of the target when it might have been easier to score.
Maza then drew a fine save from Koffi from five yards out, as the tie briefly opened up again, with both sides seemingly capable of scoring in this one.
Hicham Boudaoui really should have scored a back post header with 15 minutes of the match remaining that he steered inches off target, as Koffi made a superb save to keep out a low stinging drive from Ait Nouri.
Burkina Faso made a double change with Gustavo Sangare replaced by Cyriaque Irie as Saidou Simpore on for Steeve Yago.Minoungou hit the roof of the Algeria net as the Burkinabe continued to search for an equaliser, with time very much against them at this point.
A Traore header from close range, following a late corner, failed to trouble Zidane, as Minoungou failed in his bicycle attempt to connect to a deep Ouattara delivery.
Algeria made a double change to see out the tie but they couldn't get out of their box in the dying seconds as first Tapsoba was then wide with a swing of the boot before Simpore got a header on target but couldn't find a way past Zidane.
Algeria ultimately held on for the win to claim their place in the last 16, as Burkina Faso look at where they went wrong before going again in their final group game on Wednesday.