Steven Gerrard's side trailed in the 16th minute of the clash, Robbie Burton benefiting from a significant deflection to find the net at the Kop end.
But Curtis Jones levelled moments before the interval with a composed finish from Rafael Camacho's set-up and the 16-year-old bagged his and the Reds' second moments after the restart – via a fortunate flick off a defender.
Their lead held until the final five minutes, when substitute Bukayo Saka found the roof of the net to force extra-time and the only strike in the additional 30 minutes came from Arsenal's Tyreece John-Jules to eliminate the hosts.
Liverpool U18s: Jaros, N. Williams (Larouci, 95), Lewis, R. Williams (Sharif, 104), Boyes, Clayton, Tagseth, Dixon-Bonner, Millar (McAuley, 106), Jones, Camacho.
Unused substitutes: Glennon, Atherton.
Sorry, this video is no longer available.