"It feels absolutely fantastic and I don't think words can describe it as we are all made up," Darby told the club's official website.
"It was a real team effort and we put a lot of hard work in. City are a great side and came at us hard in the first half. We knew it was going to be tough but everyone stuck together and we came through it.
"Jack Hobbs and Godwin Antwi were rocks for us at the back along with Robbie Threlfall but this was a real team effort.
"To be honest the first game in the youth cup when we beat Cardiff we didn't really look like a good team, but as we have progressed through the rounds we have looked stronger and a better side.
"It was fantastic to have the Liverpool supporters cheering us on and they really gave us a boost, and we would like to say thank you to them for their fantastic support."