The Mersey club famously reached the third round proper in the 1991-92 season and played Premier League Tottenham two years ago.
Abraham Odoh got the visitors off to the perfect start as he hammered home from inside the box in the fifth minute.
Chris Doyle drew the seventh-tier side level with his back-post header in the 26th minute.
Sam Folarin restored the visitors’ lead three minutes later as he completed a counter-attack by skipping past Blackburn loanee goalkeeper Felix Goddard and tapped into the empty net.
Finlay Sinclair-Smith rattled the crossbar for the minnows before Lewis Thomas made a good save from Sol Solomon’s back-heeled rebound.
Solomon was frustrated again on the stroke of half-time when Matt Foulds cleared his effort off the line.
Folarin put the game to bed eight minutes after the restart as he latched onto a loose ball to add a third.
Rod McDonald volleyed home the fourth with 19 minutes to go before Stephen Dolley added the gloss at the end.