Next up for the Foxes is a Carabao Cup semi-final second leg against Aston Villa and they go into it after this 1-0 win where most of their regulars were rested.
Thomas Frank also made various changes with Brentford in Championship action against play-off rivals Nottingham Forest on Tuesday and saw his young XI struggle early on before pushing the Premier League outfit all the way.
In total 18 changes were made by both clubs – nine each – but it did not dampen the spirits of the home fans, who belted out a rendition of Hey Jude by the Beatles prior to kick-off.
Vardy was absent with a glute injury so Iheanacho replaced him up front and he put Ayoze Perez through on goal inside 20 seconds, but a superb clearance from Luka Racic stopped a certain opener with Luke Daniels rounded by Leicester’s number 17.
This was brief respite for Brentford because in the fourth minute the breakthrough occurred following a slick move.
Dennis Praet was afforded too much space in midfield and produced an exquisite crossfield ball across the floor out to James Justin and he crossed in for Iheanacho to open the scoring with his seventh goal of the season.
It could have been two for Vardy’s deputy with quarter of an hour gone, but Daniels was able to block well before Brentford created an opportunity of their own a minute later.
Jan Zamburek found Emiliano Marcondes inside the area and yet the hero of the third-round win over Stoke could only fire into the side-netting.
Brentford were boosted by the chance and produced another excellent move in the 20th minute with Dru Yearwood involved, but Halil Dervisouglu curled wide after some neat footwork and the promising opening had been wasted.
Frank’s young side, with six players aged 20 or below, stuck to their principles of playing out from the back although they were reminded of Leicester’s threat when Perez was brilliantly denied by Daniels just before half-time and Marc Albrighton headed the rebound over.
It was the fringe players who very nearly fashioned the equaliser with 62 on the clock, but the woodwork saved Leicester.
Marcondes whipped in a delightful cross, which Dervisouglu got the faintest of touches to and yet he saw his effort hit the post and Christian Fuchs managed to clear the danger at the back post.
The substitution of Iheanacho with 22 minutes left at Griffin Park further highlighted the Foxes priority this week being the Carabao Cup semi-final second leg away to Aston Villa on Tuesday.
Midfielder Kiernan Dewsbury-Hall was introduced in his place, with Rodgers eager for his side to see out the final exchanges with no last gasp scares.
He did not get his wish with Marcondes’ corner finding Racic and the defender volleying towards goal, but fortunately for the visitors Danny Ward produced a strong hand to tip the effort over.
Frank’s final roll of the dice was to bring on 11-goal man Bryan Mbeumo and he did find the net, but he was correctly ruled to be offside and Griffin Park bid farewell to the FA Cup after a 1-0 loss to Leicester, with Brentford set to move into their new stadium in the summer.