Iheanacho had scored more FA Cup goals (13) heading into Monday's (AEST) contest at Wembley – which was played in front of a small crowd – than any other player since the start of 2015-16, and it was his second-half strike which proved decisive.
Playing in its first FA Cup semi-final since 1982, Leicester had the better chances throughout, although Iheanacho's effort was the only shot on target in the entire contest.
It gave the Foxes their first win at the new Wembley in four attempts and Brendan Rodgers' side will now chase a second when they face Chelsea in next month's showdown.
The 4,000 socially distanced fans were not treated to a true opportunity until Jamie Vardy raced through after half an hour, only to slice a close-range chip into the side netting.
With Southampton having more possession without threatening Leicester's goal – they finished the first half without registering a single attempt – Wilfried Ndidi had the next chance, his looping header landing on the roof of the net.
Leicester's breakthrough arrived 10 minutes after the restart. Vardy spun his marker to get to the byline and squared for Iheanacho, who slotted in after his initial effort was blocked.
Southampton responded swiftly, mustering four shots in quick succession, with Ibrahima Diallo going agonisingly close to restoring parity with a strike which whistled just wide.
Returning from the bench after his club-enforced suspension for a breach of COVID-19 regulations, James Maddison could have made it 2-0 with 13 minutes remaining but thumped over from Iheanacho's pass.
Maddison went closer still moments later with a venomous shot which flashed across Fraser Forster's goal, though a blunt Southampton attack failed to make Leicester pay for those missed opportunities.