Town sailed through to the next round of the Vase after registering eight goals on their short trip to Coventry on a warm Saturday afternoon.
A first half hat-trick from Caine Elliott had set Valley on their way, with David Kolodynski matching that achievement after the interval, whilst Dylan Parker scored one in each half to complete the scorers.
Rugby boss Carl Adams made four changes to the side that had struggled in defeat at Wellingborough the previous weekend, with Omar Recci, Jordan Haywood, Liam Francis and Danico Johnson back into the starting eleven.
The visitors made a decent start to the game, and Elliott was soon curling one just wide of the target, with Johnson also flashing a ball across the face of the Copsewood goal shortly after.
Sean Kavanagh did have a good chance for the hosts on the break though, but he sent his volley from a dangerous position over the bar.
Town took the lead on sixteen minutes though, after Kolodynski had cleverly dummied Omar Recci's low through ball to allow Elliott to run onto it and ease the ball past Copsewood keeper Dale O'Donnell.
Elliott's and Rugby's second came nine minutes later when the midfielder dispossessed his opponent near the edge of the penalty area, before skipping past a couple of Coventry defenders to again slide one past O'Donnell.
The one-way traffic continued, and when Sam Ellis pushed Kolodynski in the back in the box, Elliott gratefully accepted the opportunity to complete his quickfire treble by coolly sending O'Donnell the wrong way from the penalty spot.
Parker made it four when his speculative shot from distance took a huge deflection off a Copsewood defender and ultimately deceived O'Donnell to loop beyond him into the top corner of the net.
The home side did rally briefly ahead of half-time though, with Ben Newey pulling off a fine save to Kavanagh's low strike and Jacob Dodds also denied by a good block by Liam Francis.
Emerging after the restart with a four goal cushion, Town then suffered a mini-wobble by conceding twice in the opening minutes of the new period.
First Alex Lock exploited some space down Rugby's left to cut in and apply a neat finish, and then Kavanagh got in behind the visitors' back-line to help the ball past Newey for Coventry's second.
Copsewood were unable to build on their momentum though, and Valley were soon reasserting themselves on the game - with sub Melu Mpande drawing a good save from O'Donnell and James Hancocks sending in a dangerous swinging free-kick just beyond the reach of his team-mates.
Town's goal rush kicked back in again on 52 minutes after Mpande had skipped past a number of defenders only to see his chance of a first goal for the senior team denied by the boot of Sam Hewitt.
However the loose ball fell nicely to Kolodynski to tap home from two yards out.
Rugby's sixth came after good interplay between Elliott and Kolodynski ended with the latter squaring the ball for Parker to fire home, and then Kolodynski's good movement and anticipation allowed to him take advantage of a loose back header by one-time Rugby man Craig Civzelis to clip the ball past the advancing O'Donnell.
O'Donnell denied Parker his hat-trick with a fine save after he had broke free down the middle, but Kolodynski completed his own with another simple close range finish when Edwin Ahenkorah's cross had looped up across the face of the goal after Mpande had done well to play him in down the left wing.
Valley Line-up: Newey; Omar Recci, Loyiso Recci, Francis; Josh Thomas, Hayward (Mpande), Hancocks (Bennett), Elliott; Parker, Kolodynski, Danico Johnson (Ahenkorah). Subs not used: Fitzharris, Marsden. Booked: None.
Coventry line-up: O'Donnell, Hewitt, Ellis, King, Civzelis, Fesel, Wassall, Dodds, Kavanagh, Lock, Mukenge. Subs: Meadows, Glaznieks, Cross, Pedley, Fraser-Walters.