Goalscoring chances were few and far between during the opening 45 minutes at St Andrews, as neither goalkeeper was called into any testing action.
However, what the first half lacked in entertainment, the second half more than made up for.
Blues striker King broke the deadlock from the penalty spot in the 50th minute after Andy King had brought him down inside the area.
Despite having a numerical disadvantage, the visiting Foxes pushed for an equaliser but Boaz Myhill stood firm in the Birmingham goal.
Their failure to draw level was eventually punished five minutes from time when King found Wood on the counter-attack and the young striker rounded Kasper Schmeichel before rolling the ball into an empty net for his eighth goal in nine matches.
The defeat was Leicester's first away from home this season.