Morton ensured new boss Jim Duffy got off to a winning start as they narrowly overcame part-time Spartans 1-0.
Cameron O'Neil's 27th-minute strike proved to be the difference, as the Lowland League champions gave a good account of themselves in torrential rain.
Andrew Barrowman missed a golden opportunity for Morton in the third minute, shooting straight at Kevin Swain having broken through on goal.
Swain then had to save from Mark Russell, who had let fly from 20 yards, and David McNeil from point-blank range either side of Michael Bruce testing Derek Gaston at the other end.
The hosts took the lead through a well-worked goal that saw O'Neil exchange passes with McNeil before firing in from 10 yards.
The second half was a fiercely-fought contest, but Spartans were unable to find a goal to take the game into extra-time despite golden chances for Michael Herd and Keith McLeod.