After Alloa had made a bright start - John Ovenstone seeing a free-kick deflected over the bar and Alex Walker having to make a vital tackle on Fraser Forrest - the home side went ahead with a slick goal.
A minute after slipping when in the clear and incurring the irritation of manager Jim McInally ("what do you have on your feet, Peter?"), Weatherson volleyed in from 12 yards after meeting a splendid Andy McLaren cross at the far post.
Constant Morton pressure followed, and Scott McLaughlin had a long-range drive deflected over by Brian McColligan and Derek Lilley's header across goal from Walker's cross found no takers.
Alloa were further bemused by two balls in their penalty after one cleared out of the ground was retrieved.
McLaren kicked fresh air and Lilley had a shot saved by James Evans before James McAlistair doubled the lead with a firm angular drive from 16 yards after McLaren's first-class pass.
Referee Hunter was replaced at half-time by assistant David Moran.
Substitute James McPake scored his first Morton goal 11 minutes after the break when he smashed the ball past Evans after McLaren's through ball was not dealt with capably by Michael Bolochoweckyj.
Alloa fought back spiritedly but 'Ton scored a fourth, McPake firing in from Kevin Finlayson's knock-on. Graeme Brown sent a glancing header from Ross Hamilton's cross past Stefan Gonet, but it was meagre consolation.