Slimani set the Africans on their way to just their second World Cup finals victory with the opening goal, quickly followed by strikes from Rafik Halliche and Abdelmoumene Djabou as the Algerians, beaten narrowly by Belgium in their opening game, raced into a 3-0 lead by half-time.
The Koreans rallied after the break with Son Heung-Min and Koo Ja-Cheol scoring, but Yacine Brahimi's calm finish in between meant there was no late drama.
A draw in their final group game against Russia in Curitiba on Thursday could be enough to send Algeria through with Korea needing to beat group leaders Belgium to stand any chance of progressing.
Algeria's Bosnian coach Vahid Halilhodzic made five changes to the team which started against the Belgians and could hardly have reaped greater reward as his side flew out of the blocks.
Sofiane Feghouli, who had scored Algeria's first World Cup finals goal for 28 years in their opening fixture, fired wide within two minutes of the kick-off and Slimani might have done better with a free header seven minutes later.
However, they did not have to wait long to take the lead when defender Carl Medjani played a 26th-minute ball over the top and Slimani squeezed between central defenders Hong Jeong-Ho and Kim Young-Gwon to lift the ball over advancing keeper Jung Sung-Ryong.
Algeria extended their lead just two minutes later when defender Halliche beat Jung to Djabou's corner to power home a header.
There was worse to come for South Korea - who did not muster a single shot on target in the first half - seven minutes before the break when Slimani's square ball presented Djabou with the chance to sweep the ball past the helpless Jung and make it 3-0.
But Hong Myung-Bo's men gave themselves hope within five minutes of the restart when Son ran on to a raking pass out of defence and turned the flat-footed Madjid Bougherra to drill a left-footed shot through keeper Rais Mbolhi's legs and into the net.
They would have been firmly back in the game on the hour had Mbolhi, who had earlier been grateful to defender Aissa Mandi for a goal-line clearance from Park Chu-Young, not pulled off a stunning save to keep out a 30-yard Ki Sung-Yueng piledriver.
However, their hopes were dashed within two minutes when Brahimi exchanged passes with Feghouli before coolly adding his name to the scoresheet.
Koo reduced the deficit once again with 18 minutes remaining when he bundled substitute Lee Keun-Ho's cross past Mbolhi, but Algeria saw out the time to claim the points.