Lewis Brown had brought the Cherries back into the contest with a well-taken lob shortly after the restart, only for captain Sammy Braybrooke to score a crisp volley to regain the Foxes' two-goal advantage.
Two late strikes from substitute Brandon Cover then built on the scoreline as City confirmed their place in the quarter-finals of the competition.
"We're delighted with that performance, and the way at times we controlled the game," City's coach reflected to LCFC.com following the match. "Ultimately, the FA Youth Cup's about getting through and we're delighted to get through to the quarter-final with a 5-1 win.
"Getting two goals in the first half, the first goal is always important in these ties. Going in at half-time, it was about playing positively and establishing control in the second half. Unfortunately we conceded, I thought Bournemouth came out in the second half with great intensity and that caused us problems early on.
"They were chasing the game and then, when it goes to 3-1, I thought the boys finished off the night with some real quality.
"We've got some outstanding individuals, but collectively as a group they've also got some real good quality. The mentality of the group's fantastic, the way they work daily is a real credit to the players and the staff.
"We're disappointed to concede, we pride ourselves on clean sheets and they're important for us. Especially early on, Bournemouth had some chances and had quite a lot of penalty box entries - more than we would like.
"Ultimately, some of the defending was good and, when we needed to be strong and resolute, we were."