The hosts earned their first league victory since January to move to within four points of safety, with Esteban Cambiasso's superb volley having initially put them ahead at the King Power Stadium.
David Nugent missed the chance to double the advantage three minutes later when he saw a penalty saved by Adrian after Leonardo Ulloa was felled by Carl Jenkinson in the area.
However, Leicester were the better side after the break and earned a crucial three points when King, who had only been on the pitch seven minutes, scored from close range in the 86th minute.
Cambiasso - who boasts UEFA Champions League, La Liga and Serie A winners' medals - stated that keeping Leicester in the Premier League would mean as much as winning a trophy and Nigel Pearson's men are now just four points adrift of 17th-placed Sunderland with eight games remaining.
Sam Allardyce's West Ham, who remain in the top half, started sluggishly and the game burst into life in the 12th minute as Leicester deservedly took the lead.
The pressure told in the 32nd minute when Alex Song's exquisite chipped pass was too high for Jeffrey Schlupp and Kouyate controlled the ball on the right of the area before finishing emphatically into the far corner.
West Ham were on the back foot and Jamie Vardy headed wide from eight yards when Ritchie De Laet's long throw was flicked on by Ulloa, before the forward fired a left-foot shot from the edge of the penalty area wide.
Leicester were nearly made to pay for their profligacy as Kouyate's low effort from an acute angle rebounded off the woodwork.
Both teams had chances to win it in a frantic ending and substitute Andrej Kramaric had a shot cleared off the line before was Leicester made the breakthrough.
West Ham failed to deal with a goalmouth scramble and King got the final touch to send the ball over the line - much to the delight of the home faithful.
