Dempsey Arlott-John opened the scoring after 31 minutes before, on the stroke of half-time, Shane Flynn doubled City's lead with a well-placed finish from the corner of the 18-yard box.
In the second period, Terell Pennant and Will Russ added to the home side's tally to set up a fifth round tie with Crystal Palace or Bolton Wanderers, who play each other on Friday.
The young Foxes, who defeated Fulham 2-1 in extra-time in the third round last month, started in industrious fashion, carving out an opening in the first 60 seconds.
Flynn galloped down the left wing, forcing Jakob Priestman out of position and crossing for Arlott-John, who nodded it wide of Sam Booth’s far post.
It was the same combination who paired up in the 12th minute with Flynn once again wreaking havoc down the channel and locating Arlott-John, with the same end result.
Leon James’ long-ranger, in the 19th minute, was the next opportunity for City to take the lead, but it flew over Booth’s crossbar.
A deserved lead eventually came Leicester’s way just after the half-hour point when another cross from the left flank - this time from skipper Luke Thomas - found Arlott-John in space.
The young striker made no mistake, ghosting past his two markers, and applying an emphatic finish into the visitors’ net to give City the advantage they so richly deserved.
Leicester were able to double it too on the cusp of half-time when Flynn expertly picked his spot in the far corner, from the left angle of Crewe’s 18-yard box.
After the break, the young Foxes retained their expressive nature and were unlucky not to register a third goal when Booth saved well from Dennis Gyamfi’s close-range effort.
It was 3-0 to the hosts with 69 minutes on the clock as Pennant twisted and turned on the right, before somehow firing into the roof of the net from a tight angle.
With seconds to spare, Thomas' free-kick is parried by Booth into the path of Russ, who converted comfortably to add another touch of gloss to the scoreline for Leicester.
The details...
City: Bosworth, D. Gyamfi, Thomas (c), James, Elewa-Ikpakwu, Kranthove, Arlott-John (Fitzhugh 70'), Ewing, Russ, Pennant, Flynn (J. Gyamfi 79')
Subs not used: Sams, Stolarczyk, Murch
Goals: Arlott-John 31’, Flynn 45’, Pennant 69', Russ 90'