Sheffield United manager Chris Wilder left the club late on Saturday and coach Paul Heckingbottom, who will take charge until the end of the season, had no time to fix the team's defensive frailty and toothless attack.
He only managed to delay the inevitable, however, and Leicester went ahead in the 39th minute when Jamie Vardy latched on to a clever pass and left Iheanacho with a simple tap-in.
Iheanacho tried to repay the favour early in the second half, teeing up Vardy for a first-time shot but Chris Basham headed his effort off the line as Vardy, who has scored only once this year, struggled again in front of goal.
Perez added the second in the 64th minute with a low shot before Vardy again set up Iheanacho for another simple finish.
He saved the best for last, firing home a powerful left-foot shot in the 78th minute to complete his first Premier League hat trick.
Vardy's shot was turned into his own net by Ethan Ampadu to complete the rout as Leicester climbed up to second spot on 56 points after 29 games, two ahead of Manchester United who play West Ham United later on Sunday.
"I don't know how to express this, I've been waiting for this day to come. I'm grateful to God, my team mates and my manager for helping me grow," Iheanacho told Sky Sports.