Charlie Huth, La'more Lee Forrester, AJ Yeguo and substitute Zac Coxon were on the scoresheet for the young Reds.
Simon Wiles' side went ahead inside the opening 10 minutes, when Huth met Erik Farkas' long ball and slipped a finish under the advancing goalkeeper.
Lee Forrester created their second with a driven delivery into the box that was accidentally turned into the net by a visiting defender.
Stockport then netted but Liverpool's lead was soon back to two after Lee Forrester's superb finish punished the 'keeper for venturing off his line.
Yeguo added the hosts' fourth before the half-time interval and a spirited response from the Hatters.
They scored three times without reply in the second period to make it 4-4 but the game was decided by Coxon's hit in minute 81.
**Liverpool:** Wright (Hilditch, 46), Owen, Cisse, Clarke, O'Connor, Upton (Garin, 58), Yeguo (Coxon, 51), Hickman (Phillips, 75), Huth, Farkas, Lee Forrester (Murray-Holme, 58).