Clayton Donaldson gave City the start they needed with a first minute goal, finishing after Andy Holt nodded down Jamie Forrester's cross.
Jamie Forrester had chance to double the lead after Dean Keates was brought down in the box, but his penalty was saved.
But Donaldson made amends by bagging his second 3 minutes later from a Simon Russell through-ball. Forrester missed a similar opportunity, so the scoreline stayed at 2-0 at the break.
Forrester finally found the net in the second half after a defensive blunder, and Russell completed the scoring at 4-0 from Forrester's back-heeled assist.
City line-up: Musselwhite; Regan, Whittle, Anderson (Byron 77), Peat; Russell, Fry, Keates, Holt (O'Neil 75); Donaldson, Forrester (Cooper 81).
Un-used subs: Harvey, Tomlinson.