Up to the interval, Stranraer had looked the more settled side and had made David McGurn the busier of the two keepers.
Not that there were many clear-cut chances going, but the home side definitely had the bulk of them and McGurn was twice was in trouble.
First with a David Hamilton header which clipped the bar and had to be hurriedly cleared and then when the Morton keeper dropped the ball and it was cleared for a corner.
It was a different tale after 57 minutes when Morton opened the scoring.
Paul McGowan did the lead-up work but then sent his shot wide only to see it strike Stranraer's Lee Sharp and finish up in the other side of the net.
Stranraer's confidence then deserted them completely and Morton suddenly found themselves with all the space they needed in which to run freely and to such an extent that they doubled their lead four minutes later.
Stevenson got the goal his earlier play deserved and then after 81 minutes when another of his drives hit the foot of the post, Chris Millar had only to steer home the rebound to round off a good win.