An Adam Hammill penalty gave Liverpool the lead in the second half when Hammill himself was brought down in the box. Miki Roque, who usually plays for the reserves, added the second after putting in a rebound from one of his own shots late on.
Coach John Owens said: "We had had a very good game earlier this year at Stoke and just won 1-0, but it was one of our hardest games. We were prepared for the same again. It was very tight and stayed at 0-0 up to the break with both goalkeepers playing well.
"But we took the lead from the spot and added a second through one of the Melwood lads Miki Roque.
"We had five of the lads from Melwood playing at second year scholarship age to give those guys a match with a view to next week's Youth Cup match.
"Hopefully we can build on that and see an improvement."
Liverpool slipped back down to third in the table on Saturday after Man Utd and Blackburn both picked up points.