From the start Morton went all out to impress new manager, Jim McInally. Alex Williams had an early shot blocked and a Chris Millar effort was deflected on to a post by David Murie.
The pressure was maintained, Gary O'Connor saving a low John Maisano drive and then palming over a Stewart Greacen header from Maisano's corner.
There was no easing up from the home side, and Maisano almost broke the resistance when he forced his way into the box following an attractive four-man move, but O'Connor dashed from his line to make a brave save.
James McAlistair's low drive was tipped round the post by O'Connor but, just after the half-hour, Gareth Hutchison passed up a wonderful chance to put the visitors ahead when his free header from Paul Hampshire's free-kick was wildly high.
It was McAlistair who made the opener. He beat David Murie in a challenge before reaching the goal-line and delivering a perfect cross for Jason Walker to head past the helpless O'Connor at the far post.
The Borderers came into affairs more in the second half and were unlucky when a Mark Cowan header from a Martin Neil cross was kicked off the line by Marco Maisano, but they were finally deflated when the heroic O'Connor felled substitute Paul Walker in the box two minutes from time and was comprehensively beaten by John Maisano's spot-kick.