Algeria led 3-0 at halftime, but withstood a stronger South Korean second half performance to claim its first World Cup win since 1982 and move into second place in Group H with one match left to play.
Defeat for South Korea means it must now beat already-qualified Belgium to stand a chance of progressing to the knockout stages. Belgium leads with six points, Algeria now has three, while Russia and the South Koreans have one apiece.
Slimani, one of three Algeria scorers who didn't start in the opening 2-1 loss to Belgium, broke the deadlock in the 26th minute with a fine solo goal, running onto a through ball between two defenders and then touching it past Jung Sung-ryong.
Two minutes after the restart, defender Rafik Halliche powerfully headed in a corner after Jung came off his line but failed to make contact with the ball.
Algeria added a third goal in the 38th when Abdelmoumene Djabou side-footed home from close to the penalty spot after Slimani pulled the ball back for him.
South Korea didn't manage a shot on target all half, and struggled to break down Algeria's attacks.
However, it was far more positive after the break and pulled a goal back in the 48th minute when Son Heung-min shot through the legs of Rais Mbolhi.
Hopes of a remarkable comeback were soon extinguished, though, when swift passing by Slimani put Yacini Brahimi through on a counter attack to make it 4-1.
South Korea captain Koo Ja-cheol bundled the ball home on 72 minutes from close range, but Algeria's defence held on to secure a historic victory.