The visitors were fortunate to be level at half-time after Bob McHugh had a goal disallowed but they produced a spirited performance after the break to secure the win.
They made the breakthrough after Gregor Buchanan fouled Dario Zanetta in the box and Alan Trouten stroked the penalty into the bottom left corner.
The Wasps were awarded a more dubious spot-kick 11 minutes later as Reghan Tumilty looked to have hauled down substitute Jake Hastie outside the area. However, referee Colin Steven pointed to the spot and Iain Flannigan converted.
Andy Graham’s goal-line clearance had denied Morton an early lead and typified the resolute defending of the visitors.
Morton’s best chance fell to Morton captain Jim McAllister whose left-footed strike was well saved by Neil Parry following a surging run and pass from Gregor Buchanan.