Against FA Women's Super League opposition, a youthful Foxes side in fact trailed for 79 minutes in Quorn following Brooke Chaplen's penalty for the Royals late in the first half.
However, City fought back admirably with a 25-yard effort from Bailey-Gayle, in off the post, to send a dramatic tie to a penalty shootout.
The clash was settled in extra-time with Bailey-Gayle, 18, netting about stunning effort, which cannoned in off the crossbar, to send Leicester into the last-eight stage.
An upturn in recent form has seen the Foxes climb to sixth in the FA Women's Championship, winning four games in a row in all competitions.
Following the success over Reading, Leicester City Women manager Johnathan Morgan labelled it the 'best victory' he had ever been involved with, detailing his pride in the squad's performance.
A date for the quarter-final tie against Manchester City - who currently sit atop of the Women's Super League standings - will be set and announced shortly.
The details...
City: Lambourne, N. Johnson, Popple (Brown 66'), Cataldo, Devlin, Blanchard, James (Bailey-Gayle 61'), Thomas, Rutherford, Plumbtre, L. Johnson (Everett 94')
Subs not used: Field, Smith, Lawrence
Goals: Bailey-Gayle 79', 114'