Mussels’ joint boss Mark Burt attributed his side’s slow start to rustiness having not played for two and a half weeks, whereas Rebels faced Whitehawk on Saturday having not been affected by the weather.
Rebels fielded line-ups with a mixture of youth and experience, but straight away the one division gulf between the sides was evident as current-holders Worthing took the lead after just 54 seconds.
Ex-Shoreham stalwart Matt Hurley played in Mike Huckett, who skipped past two midfielders and slotted home past onrushing keeper Sam Armfield.
Six minutes later, the tie seemed over as a contest as Ryan McBride volleyed home Glenn Woodburn’s cross after Tom Manton’s pass had picked him out.
Youth teamer John Newman had an effort pushed away by Armfield, but having weathered the early storm Mussels began to show their mettle.
They should have been given a way back into the tie when Richard Hudson was sent tumbling under Luke Gedling’s challenge, but referee Ian Moore waved away their appeals.
Shoreham had gained a foothold and they finally tested keeper Alan Mansfield on the half-hour when he was forced to tip away a Matt Beeston header.
Mussels continued on the front foot after the break and almost reduced the arrears through Gareth Green who thumped the crossbar with a free-kick.
After both sides traded half-chances, the next goal in the complexion of the tie would be crucial, but Worthing got it and with it sealed their progress into the last eight.
Adam Hunt, dual signed from Lancing, drilled the ball across and with several players queuing up to score, Tom Manton bundled in from close range to round off the scoring.
That was the final noteworthy action of an entertaining cup-tie and, while not high on the list of priorities for either, it was a useful run-out for the Shoreham squad and a taste of first-team action for some of Worthing’s fringe and youth players.
Rebels boss Chris White said: “We played well in spells, at times we were sloppy and took our foot of the pedal and although this cup doesn’t mean a lot to most teams we won it last year so we’re in it to win it.
“We had a few out so gave some of the younger players their chance, we did enough and it was something a bit different for us, the league is obviously our priority as it is for them too, but a cup run is always nice.”
WORTHING: Mansfield; Hunt, Hurley, O’Regan, Woodburn; Huckett, Budd, Gedling, Manton; Newman, McBride. Subs: Metcalf, Clack (Newman, Woodburn 45), Christie (Budd 70), Piper, Bardaleau.