Richarlison had given the visitors a shock lead at the King Power as Everton produced a performance that may do enough to save manager Marco Silva's job ahead of the Merseyside derby on Wednesday.
But the introduction of Iheanacho just after the hour mark changed the game as the Nigerian set up Jamie Vardy to equalise and then dramatically scored the winner, which was initially ruled out for offside before a VAR review overturned the decision.
A sixth straight Premier League win sends Brendan Rodgers's men three points clear of defending champions Manchester City in second.
Silva was reportedly on the verge of losing his job earlier in the week after a run of just two wins in 10 league games that has left the Toffees hovering just above the relegation zone.
The Portuguese responded with a change of formation and his switch to a 3-5-2 worked to perfection for the opening goal.
Djibril Sidibe broke upfield from right wing-back and his cross was bulleted home by Richarlison with a diving header.
Leicester had already seen two half-hearted appeals for a penalty waived away for challenges on Ayoze Perez by the time that referee Graham Scott did point to the spot for Mason Holgate's apparent trip on Ben Chilwell.
However, a lengthy VAR review concluded no contact had been made and the Foxes were frustrated once more.
Leicester laboured in the search for an equaliser, but the introduction of Iheanacho as a substitute just after the hour mark provided the much-needed spark.
His first effort was too close to Jordan Pickford, but he got in behind the Everton defence again moments later and squared for Vardy for score his 13th goal of the season at the back post.
Vardy had been quiet for the first 68 minutes, but came alive in the final quarter.
James Maddison should have done better from his cross as his finish was straight at Pickford before Vardy rose highest in the box but just planted his header from Youri Tielemans's cross too high.
Just as Leicester were turning the screw, Everton were inches away from retaking the lead in spectacular fashion as substitute Moise Kean nearly scored his first goal for the club with a curling effort that hit the side-netting.
Silva's men appeared to have weathered the Leicester storm, but deep into added time Iheanacho collected Ricardo Pereira's pass, turned inside and found the bottom corner.
A nervous wait ensued with the linesman's flag raised, but a VAR check ruled the former Manchester City man onside for his first league goal since September 2018.