Lucciano Becchio made no mistake with the penalty for his 13th goal of the season.
Leicester struggled to gain possession in the early stages but forced Paddy Kenny to make a save when Richie De Laet shot from an acute angle.
Nigel Pearson made two half-time changes which nearly provided an instant reward when Anthony Knockaert came close with a low shot.
Leeds came close to making the game safe minutes later through Michael Brown, who drilled a shot inches past the post.
The Foxes failed to rally in the closing stages and missed out on the chance to improve their playoff position.