Jayden Joseph fired Russell Martin's men in front just past the hour mark with a rare flash of inspiration from the visitors.
However, Grant slid home from close range to restore parity after fellow substitute Emile Acquah's header had rebounded off a post.
County looked by far the better of the two teams during the early skirmishes, with Cal Roberts firing just over the crossbar and Ollie Cooper hitting the woodwork after charging down Alex McCarthy's clearance.
The first half was bookended by periods of pressure from the hosts as James Gibbons, Luka Smyth and Roberts tested the Foxes' defensive resolve.
Bobby De Cordova-Reid had been the visitors'stand-out performer and, moments after the interval, he saw a goalbound effort hacked off the line by Nick Tsaroulla.
It was Joseph who handed Leicester a 62nd-minute lead when he unleashed a powerful right footed drive from Sammy Braybrooke's lay-off.
But, 19 minutes later, Grant finished in clinical fashion after Acquah had been denied by the upright.
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