Hume equalised from the penalty spot after Tom Soares had given Palace the lead.
Leicester carved out the first clear chance of the game on eight minutes when Elvis Hammond shot wide, then four minutes later Gabor Kiraly pulled off a superb reaction save to keep out a powerful header from Josh Low.
But it was Palace that took the lead in the 16th minute when Leon Cort's header from a Gary Borrowdale corner ricocheted off the bar with Soares on hand to crash home the rebound from eight yards.
A moment later Leicester were fortunate not to fall further behind when Soares turned provider for Cort to head against Conrad Logan's left-hand post.
City responded with Hume firing wide from a tight angle and Low's cross which landed on the roof of the net, but it was the visitors that remained in control as the game reached the half-hour mark.
But the encounter was all-square on 37 minutes when Hume scored from the penalty spot for his fifth goal of the season. Stephen Hughes played the ball forward to the feet of Hammond who was held back by Darren Ward as he tried to turn for goal.
The Palace defender was shown a yellow card and Hume stepped up to score low to Kiraly's left.
The rest of the half remained an even contest and it was much the same story after the break as both sides struggled to mount a serious threat in what became an increasingly scrappy affair.
Palace did create a chance on 64 minutes but Clinton Morrison failed to exploit Matt Lawrence's accurate cross as he headed tamely into the keeper's arms.
In the 77th minute the Londoners were wasteful again when Logan got down well to palm away a Morrison effort only for the ball to run into the path of substitute Stuart Green who, from a good position, fired wide of the goal.
Leicester hit back with a run and shot just beyond an upright from substitute Momo Sylla before Richard Stearman and Danny Tiatto had shots which failed to hit the target.
Then, in the final minute, Palace squandered a fine opportunity when another of their substitutes, Dougie Freedman, bore down on the Leicester goal but Logan stood his ground to make a fine save.