The visitors, reduced to 10 men by Rowan Vine's red card, came a poor second to a Cowdenbeath side they now trail by 16 points with just seven games remaining.
While Cowdenbeath now have the teams above in their sights, Morton are destined to miss out on next season's lucrative Championship featuring Rangers and Hearts.
The hosts made the ideal start with a goal inside the first six minutes, Rory McKeown swinging the ball into the box where Kyle Miller rose to head high behind a helpless Derek Gaston.
Morton responded with some spirit in a bid to draw level. Thomas O'Ware went close with a shot which skidded wide before a series of corners established a pattern of pressure on the home goal.
Morton gave the travelling fans in a meagre 318 crowd some hope when Craig Knight's shot gave home keeper Sebastian Usai his first serious action of the game before Garry O'Connor sent a header just over.
The visitors' hopes of clawing their way back into contention suffered a major setback early in the second half when Vine was shown a straight red after a bust-up with a linesman.
And the game was up for Morton in 63 minutes when Greg Stewart pounced on James Fowler's through ball and calmly stroked the ball beyond Gaston.
The home side were now in total control and they made the extra man count again with a patient passing move which culminated in Stewart setting up Miller for his second goal of the game.