The Welsh outfit took a fourth minute lead courtesy of a spectacular strike from Alex Titchener, but Toni Gomes levelled for Neil Critchley's side.
Both teams then had opportunities before the young Reds were awarded a penalty six minutes from time, and Brannagan held his nerve to seal the victory.
Reflecting on the win, Critchley told Liverpoolfc.com: "It was a good work out for us and it was great to get the first game under our belts.
"Rhyl were great hosts and there was a good crowd who saw a decent game.
"We have only been back training for a week so it was good to step up our fitness and there were some good performances."
Liverpool U23s: George, Whelan, Johnston, Jones, Masterson, Phillips, Coyle, Virtue, Brannagan, Gomes, Dhanda. Subs all used: Firth, Williams, Parker, Alves, Kane, Camacho, Simmonds, Millar.