A second-half header has made history for Vahid Halilhodzic's side, who reach the knockout stage for the first time
An early header from Alexandr Kokorin gave Fabio Capello's men the initiative, but Islam Slimani struck the decisive blow in the second period as Igor Akinfeev failed to make contact with a cross into the box.
Famously eliminated at the finals in Spain 32 years ago when West Germany and Austria played out a mutually beneficial 1-0 scoreline to send the north African side out, Algeria put the record straight in Curitiba on Thursday.
Coming into the Group H clash, Algeria knew a draw would likely be enough to seal qualification - barring a heavy victory for South Korea over Belgium - whereas Russia required a win to advance.
That result looked to be on the cards when Fabio Capello's men went in front as early as the sixth minute, as Aleksandr Kokorin headed home Dmitri Kombarov's pin-point delivery.
Russia looked comfortable with their advantage until the hour mark, when Igor Akinfeev - who made a horrible mistake in conceding during the 1-1 draw with South Korea - failed to make a connection with Yacine Brahimi's set-piece delivery and Islam Slimani converted from six yards.
Having qualified in second place behind Belgium - who beat South Korea 1-0 - Algeria will now take on Germany in Porto Alegre on Monday in the last 16.
Both sides made one change from their previous outing, Essaid Belkalem replacing Madjid Bougherra for Algeria while Capello called up Aleksandr Kerzhakov in place of Maxim Kanunnikov.
Kokorin, whose place in the starting XI had been the subject of discussion in the build-up to the game, gave Russia a dream start when he rose to head Kombarov's perfect left-wing cross into the top corner.
Russia continued to have the better of the play and Oleg Shatov went close in the 26th minute with a swerving drive from distance that drifted narrowly wide of the right-hand post.
Algeria tested Akinfeev for the first time in the 30th minute, the goalkeeper flying high to his left to tip Slimani's header over the crossbar.
Slimani tried his luck with another header two minutes before the interval, but this time his effort lacked power and was comfortably held by Akinfeev.
Aleksandr Samedov went close to doubling Russia's tally shortly after the resumption, but Rais M'Bolhi raced from his line to deny the midfielder.
That save proved crucial as, on the hour-mark, Akinfeev came to palm away Brahimi's free-kick but was beaten to the ball by Slimani, who nodded into an empty net.
Kerzhakov tested M'Bolhi 10 minutes later as Russia pressed to regain the lead, but the keeper got down well to his right to smother the well-struck shot.
Although Russia saw a lot of the ball in the final stages, they were unable to breakdown a stubborn defensive line as Algeria celebrated a landmark achievement.