Valley's Emirates FA Cup run was ended by Barwell at Butlin Road as the visitors navigated a straightforward passage into the next round with a goal in each half.
The visitors underlined their higher-level status with a good all-round performance and two well-taken strikes from Brady Hickey and Beck-Ray Enoru, with Rugby rarely posing a goal threat of their own.
Dan Summerfield replaced Josh Thomas at left wing-back in the only change to Town's starting line-up from the battling Bank Holiday Monday home win over Harborough, but it was the away side that took the initiative from the off.
The opening two chances of the game fell to Hickey from similar positions on the right-hand channel, when first he shot on the turn to draw a decent save by Matt Hill and then from a slightly tighter angle he put in a weaker effort which again was taken low down by Hill.
Jahvan Davidson-Miller's rising cross-shot was also parried away by Hill, with Thomas Cursons then seeing his strike ruled out after he slid the ball home from an offside position.
Rico Taylor's rolled a long-range free-kick to give Canaries stopper Harry Griffiths an easy take in Valley's only shot of the half, before the inevitable happened on 41 minutes when Hickey's downward header to Cursons' perfectly-flighted cross beat Hill and opened the scoring in the process.
Davidson-Miller's low strike almost doubled Barwell's tally ahead of the interval, but the away supporters did not have to wait long for a second after the restart, with Enrou finishing things off with a powerful strike into the bottom corner after Hill had pushed away an initial Hickey shot and Davidson-Miller had seen his follow-up effort blocked.
The visitors threatened to take the game away from Rugby, with Cursons missing out when he failed to connect properly with a volley at close quarters and Davidson-Miller also slashing one wide from an equally well-placed position.
In between these two misses, Barwell had had another goal struck off when Hickey was also adjudged to be offside in a more marginal decision this time, with Town finally threatening Griffiths's goal when sub Jordan Wilson showed good strength to run onto a pass from fellow replacement Karabo Motshweni to then fire one at the keeper.
Hill kept out Cursons' strike from the edge of the box and Hickey blazed high of the target after he jinked his way past a couple of Rugby defenders, before the hosts ended the game well with two decent chances that saw Kyle Burke's long-ranger drift narrowly wide of the post and Dan Summerfield fire one into the side-netting after a pacy run down the right wing.
Valley
Line-up: Hill; Rosser, Burke, Francis (Ahenkorah); Recci, Summerfield, Lee, Fitzharris (Motshweni), Taylor; McDonald (Wilson), Sahota. Subs not used: Lock, English, Kear, Thomas. Bookings: Burke, Fitzharris.
