The Seasiders struck three times in the second half at Mount Pleasant to eliminate Shields at the second qualifying round stage of the competition.
The Mariners were without their usual wing-back pairing of Jordan Hunter and Blair Adams, who both missed out with injuries, while Dan Martin made his debut a day after joining on loan from Accrington Stanley.
It was Marske – who defeated Shields 3-2 in pre-season – who had the better early opportunities, though, with a series of efforts from just outside the area by Glen Butterworth offering an indication of their threat.
Both sides were mainly restricted to strikes from distance in the opening exchanges, and the closest the Mariners came was a sweet hit by Martin which cleared the crossbar.
There were signs of encouragement for Shields as the first half progressed, with Darius Osei sparking them into life with a rampaging run from the halfway line before firing over when he advanced into a shooting position.
Dillon Morse had a more presentable opportunity as Robert Briggs picked him out with a free-kick from the right, but his header was straight at home goalkeeper Ryan Catterick.
Conor Tee powered over from distance and there was another demonstration of Osei’s quality as he skipped past his man on the left and stabbed a low effort which Catterick did well to save with his feet.
The sides went in at the break goalless but Marske emerged from it the stronger, and after Adam Boyes had volleyed not far over within five minutes of the restart, the breakthrough soon arrived.
Almost inevitably, the in-form Boyes was the player who provided it as he latched onto a straight ball over the top and used his strength before producing a precise finish into the bottom corner with his left foot.
Shields had two golden chances to equalise in a key period of the game, but Morse’s header was cleared off the line and Osei was denied by an excellent block after being played in on goal by substitute Will Jenkins.
The Mariners were punished for not taking those opportunities when Curtis Round doubled Marske’s lead 11 minutes from the end, brilliantly chipping the ball home from the left of the area following a pass by Boyes.
Craig Gott completed the scoring with a crisp finish into the bottom corner from 20 yards as Shields pushed for a late route back into the game, and although that as good as killed the tie, the away side did have late opportunities through Jenkins and Alex Kempster, with the former going closest as he forced Catterick into a fine stop with his feet.
Graham Fenton’s side now have a week to prepare for a key meeting with Warrington Town in the league on Saturday.
South Shields: Boney, Martin, Bodenham, Shaw, Morse, Ross (c), Osei, Briggs, Massanka (Hodgson 77), Tee (Jenkins 71), Kempster. Subs not used: Steavens, Agnew, Neary, McGowan, Rossiter.