Last year's champion Shahjalal and runner-up Jibon Boli did not participate in this year's competition
The hosts, looking to cement their top-four status, were cruising when goals by David Silva and Gabriel Jesus put them 2-0 ahead after 36 minutes but Leicester hit back strongly.
Shinji Okazaki smashed in a sensational volley before halftime and the hosts had a huge let-off in the 77th minute when Riyad Mahrez's penalty was ruled out because he inadvertently made a double contact while striking the ball.
Jamie Vardy also missed a late chance but a nervy City hung on to move to 72 points with two games remaining.
Liverpool have 70 points with two games left while fifth-placed Arsenal have 66 with three to play.