Iheanacho gave the home side the lead in the 39th minute after latching onto a perfect cross from Jamie Vardy. The two forwards linked up incredibly throughout the game, with both players contributing to the Leicester goals.
Perez marked his return from injury with a stunning long-range strike in the 64th minute, whilst Iheanacho completed his hat-trick with two more quality finishes in the second half.
United's woes continued as Ethan Ampadu scored an own-goal in the 80th minute after connecting with Vardy's powerful strike.
The story before the game
The main talking point before the match was the recent departure of former Sheffield United boss Chris Wilder.
Wilder left the club by mutual consent earlier this week, having endured a disappointing season, leaving United bottom of the Premier League and destined for relegation. Former Leeds United boss Paul Heckingbottom took charge as interim boss until the end of the season.
The Foxes were boosted by the return of Perez and Jonny Evans following injuries - both players came straight back into the starting eleven.
Rodgers made three changes to the team that beat Brighton & Hove Albion with Daniel Amartey, who scored the winner against the Seagulls, dropping to the bench as Evans returned to defence.
Heckingbottom also made three changes to the Sheffield United squad that lost 2-0 to Southampton. Chris Basham, Oliver Burke, and captain Billy Sharp returned to the starting line-up to face Leicester.
Early chances for Leicester
The Foxes dominated the game quickly, with Vardy almost putting the home side ahead in the eighth minute, his low-driven strike was fired wide of the post as the Blades defended frantically.
Leicester nearly scored just minutes later as Perez met Timothy Castagne's cross to aim a clever header towards goal, the ball just bounced off Aaron Ramsdale's far post and away from danger in the 12th minute.
Vardy once again came close to putting the Foxes ahead, catching Kean Bryan in possession and smashing a close-range shot straight at Ramsdale's legs in the 18th minute.
Foxes draw first-blood
Leicester finally broke the deadlock in the 39th minute through Iheanacho. The Nigerian forward fired a close-range effort past Ramsdale after Vardy delivered a perfect low cross for his strike partner.
With the Foxes leading at half-time, Rodgers made one change as Marc Albrighton replaced Ricardo Pereira on the right-wing.
Vardy and Iheanacho continued to terrorise the United defence in the second half. The Nigerian set up Vardy for a close-range effort in the 50th minute but was denied by Basham's goal-line clearance.
The two forwards once again linked up in the 59th minute, as Iheanacho played in an inch-perfect through ball to Vardy who drilled his shot straight at Ramsdale, keeping the score at 1-0.
Leicester turn on the quality
The Foxes went 2-0 up after an outstanding long-range effort from the returning Perez. The striker thundered a powerful effort past a stretching Ramsdale in the 64th minute.
Leicester soon scored their third after an incredible combination from the two strikers once again. Vardy played a beautiful curling through ball to the Nigerian who fired his effort past a helpless Ramsdale in the 69th minute.
Whilst the first three goals were good, Iheanacho saved his best for last. The forward scored his first Premier League hat-trick after firing a long-range effort into the bottom corner of Sheffield United's goal in the 78th minute.
Vardy once again had a hand in United's misery in the 80th minute. His powerful, low strike deflected off Ampadu and into the Blades' goal. The forwards' effort was heading off-target and had to be counted as an own-goal.
As the final whistle blew, the Foxes finished the game with a remarkable 5-0 victory.
Iheanacho can step up for Leicester
Whilst Vardy has been the Foxes' first-team striker for many years, Iheanacho once again showed his quality with an outstanding hat-trick. The 24-year-old has been a bit-part player during his time at Leicester but this could be his chance to become one of the first names on Rodgers' team sheet.
Foxes have the depth to create goals
With James Maddison and Harvey Barnes both out injured, many Leicester fans would have been worried about the creativity in the Foxes attack. Barnes has been a revelation going forward for Rodgers' side whilst Maddison has added goals to the team which were desperately needed.
However, with Youri Tielemans producing high-quality performances and Perez stepping up with goal contributions, Leicester may still be able to challenge the top teams despite missing some key players for the final fixtures of the season. The assists from Vardy and link-up play with Iheanacho will also have Leicester fans excited, as it relieves some of the goal-scoring pressure from Vardy.
Confidence heading into the FA Cup
After their 2-1 win over Brighton and producing a masterclass in goal-scoring against Sheffield United, Leicester will now look ahead to their next match against Manchester United in the FA Cup Quarter-Finals.
Rodgers will now have several options when choosing his side for the big fixture as his players will want to show that they can step up to the challenge when needed.