Morton earned their place in the second round of the Scottish Cup after coming from behind to win comfortably.
The Shire took a surprise lead after nine minutes when Scott Livingstone shot through a crowded goalmouth.
Alex Williams knocked in the equaliser with quarter of an hour gone before Peter Weatherson's deflected shot after 32 minutes put the hosts in front.
Two minutes later, Chris Millar connected with Paul Walker's pin-point cross and buried his header.
Morton: McGurn, Collins, Gilbride (Jason Walker 66), Greacen, Keenan, Millar, Adam, Paul Walker (John Maisano 66), McAlistair, Williams (McClean 77), Weatherson. Subs Not Used: Marco Maisano, Coyle.
Booked: Millar. Goals: Williams 14, Weatherson 32, Millar 34.
East Stirlingshire: Mitchell, Harvey (Peutherer 75), Walker, Livingstone, Baldwin, McGhee, Ross, McAuley, Ure, Donaldson (Findley 88), Dunbar. Subs Not Used: Denham, Parks, Gilpin.
Booked: Walker, McGhee, Baldwin. Goals: Livingstone 9.
Ref: W Collum.