Greenock are within touching distance of league leaders Dunfermline Athletic with nine games gone and move into third place, Andrew Barrowman's strike just after the half-hour mark proving enough to seal the points.
The first real chance of the afternoon came the way of the visitors, with Albion's Steven Doris's right-footed shot from the centre of the box flying high and wide to the right of the goal 20 minutes in.
Greenock's Stefan Milojevic nearly opened the scoring with a dangerous header from the right side of the Stirling box just going wide on 29 minutes, but Barrowman would not make the same mistake moments later.
After some good work from Mark Russell, Barrowman let fly from the left side of the six-yard box to the bottom left corner of Reidford's goal.
Barrowman picked up an injury after the break as the minutes began to tick down and the game became scrappy, David McNeil replacing the goalscorer and making an instant impact.
After receiving the ball from Derek Gaston, McNeil shot through bodies and saw his shot hit the back of the net, doubling Morton's lead and making safe all three points.