With the match to be played on the hosts' 3G astroturf pitch, the decision was taken not to risk recovering strike duo Peter MacDonald and Archie Campbell.
Impressive left-sided midfielder Aidan Fulton fired Ton into the lead on 19 minutes when he crashed a ferocious, angled drive past home goalkeeper Blair Lochhead at his near post.
Substitute Dylan McLaughlin sealed the win on 74 minutes, prodding home with his first touch after Thomas O'Ware, pictured, had headed down a Kyle Wilkie corner.
The Spiders ensured this was a full-blooded affair and were punished for their strong-arm approach in injury time when Lawrence Shankland was shown a second yellow for a foul on Declan McDaid.
Morton (4-1-4-1): Hutton (6); Boyd (7), Naismith (7), O'Ware (7), Halsman (6); Wilkie (7); Wallace (6) (McDaid 58, 5), McLean (6) (Keogh 46, 6), Moss (5), Fulton (7) (D. McLaughlin 72, 5); Hawke (6).
Subs not used: Frizzell, Jamieson.