The west African nation failed to make the most of their 3-2 first-leg lead in their qualifying play-off and paid the price for not showing more ambition.
Algeria captain Madjid Bougherra scored the crucial goal just after half-time which sent his nation through to their fourth World Cup.
The game was short on chances and neither side appeared to want to risk too much with such a big prize at stake but Algeria, unbeaten in Mustapha Tchaker Stadium since 2002, quickly assumed the upper hand.
However, they failed to create much and their only chance of the first half saw Faouzi Ghoulam's left-wing cross headed wide at the near post by Islam Slimani.
Even Bougherra's goal four minutes after the interval was courtesy of a huge slice of luck.
The debate as to whether the former Rangers centre-back should still have been on the pitch after receiving only a booking for launching a two-footed, off-the-ground lunge at Charles Kabore just before the break was still continuing when the goal went in.
That El Arabi Soudani had already escaped without punishment for planting his boot into the chest of Jonathan Zongo as he tried to win the ball was another source of irritation for Burkina Faso.
That feeling of frustration grew when Ghoulam's free-kick dropped to Bougherra, now plying his trade with Qatari side Lekhwiya, and while his shot was blocked Bakary Kone's goalline clearance rebounded off the Algeria captain and back into the net.
Burkina Faso, who registered five changes from their first-leg victory, came to life in the final quarter of the game but substitute Alain Traore's off-target drive was the closest they came to troubling goalkeeper Mohamed Zemmamouche.
In the first of four added minutes - of which only two were played - at the end of a half punctuated by endless time-wasting, Algeria almost pressed the self-destruct button when a corner was deflected on to their crossbar.
However, Burkina Faso, who were 92nd in the world rankings earlier this year prior to finishing runners-up in the African Nations Cup, ultimately went out with a whimper.