The young Foxes were the hosts in Seagrave as Wanya Marçal appeared for the Club's Under-21s side after impressing for the senior outfit at the start of the 2023/24 campaign.
The Portuguese forward, 20, opened the scoring for City on the half-hour too, although the Clarets were level at the break as Joe Westler restored parity.
While Amani Richards initially got Leicester back in front, the game started to get away from the home side in the second half. Basilio Rieno Socoliche and Lewis Richardson had Burnley in front by the 64th minute.
Westley's second of the night extended their lead too, although Silko Thomas did get one back for City late on. They're third in Group G after one game, a pool they share with Fulham and Brentford.
Leicester next face Wigan Athletic's senior side away in the EFL Trophy group stage on Tuesday (7pm BST kick-off).
The details
City: Doherty, Cover, Nelson (Appiah 46'), Wilson-Brown, Ali (Hill 65'), Amartey, Raikhy (Aluko 65'), Thomas, Richards (Pennant 72'), Marçal, Maswanhise.
Subs not used: Young.
Goals: Marçal 31', Richards 56', Thomas 90+4'