The newly-promoted Addicks drew their opening game of the season to Birmingham on Saturday but put themselves on course for all three points with a first-half double.
Last season's top scorer Bradley Wright-Phillips opened the scoring on 18 minutes with a low strike from the edge of the area.
Leicester, who had earlier wasted the best chance of the match when Lloyd Dyer fired over from close range, started the second period well as they attempted to get back into the game.
Nugent's pass found Ben Marshall on the City left and King was presented with an easy finish from the former Stoke City wingers' cross.
The visitors pushed for an equaliser but Nugent was guilty of missing a one-on-one late in the proceedings as Charlton hung on to earn three points.