And things went to plan when the visitors took the lead on 18 minutes. Paul Dickov played in Alan Rogers, who shot through Glennon.
However, City deservedly pulled back an equaliser just after the half-hour mark. Gary Alexander got his first of the season from Phil Jevons' pass, shooting into the top corner of Ian Walker's goal.
But it took only 16 seconds for Leicester to get back in front, with Paul Dickov scoring after coming forward from the kick off.
Rogers with his second and James Scowcroft added to their lead before the half time mark of extra time to dash City's hopes of a giantkilling feat.
Ian Ashbee headed in a consolation goal two minutes from time to make the result look more respectable, but City will still have felt hard done by to concede 4 and go out.
City line-up (4-4-2): Glennon; Regan, Whittle, Anderson, Smith; Appleby (Johnson 66), Ashbee, Keates (Price 96), Williams; Alexander, Jevons.
Un-used subs: Musselwhite, Edwards, Bradshaw.