Raheem Sterling made it 1-0 in the fourth minute after a jinking run.
Daniel Sturridge added a second after a brilliant pass from Steven Gerrard.
Yoann Arquinn's header gets Notts back into it.
Adam Coombes equalises late on to send the game to extra time.
Sturridge's second restores Liverpool's lead just before half time in extra time.
Jordan Henderson made it 4-2 with this excellent run and finish.