Peter Weatherson did his best to spark the home support, testing Myles Hogarth with a low drive which the keeper did well to save, given that the ball took a slight deflection.
Stewart Greacen then danced his way into the Albion box but the big defender's imitation of a dazzling winger ended with a poor shot into the side-netting. Weatherson tested Hogarth again with a low drive similar to his previous effort but the visiting number one was equal to the task.
Meanwhile, Albion carried no attacking threat but David O'Brien's cross, which just eluded the head of Robert Dunn, forced Ryan Harding to head the ball over his own bar.
Weatherson had a third shot saved but David McGurn in the 'Ton goal had to sprint off his line to beat Paddy Connolly to a throughball.
The second period produced more of the same, both sides restricting one another to unproductive possession some distance from goal.
After a flurry of substitutions failed to bring around any improvements in the level of entertainment, Morton scored a spectacular winner eight minutes from time.
Scott McLaughlin took an inswinging corner from the right and it bounced once before Greacen, with his back to goal, swivelled athletically to send an unstoppable bicycle kick past Hogarth.